Wikiversity:Colloquium

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== mediawiki2latex ==

Hi,
[https://mediawiki2latex.wmflabs.org/ mediawiki2latex] exports Wikiversity to  Pdf, Epub, Odt and LaTeX.
I suggest to add a new link to the tools in the in the section Tools.
You may try this out yourself just now by copying  User:Dirk_Hünniger/common.js to common.js in your user namespace or by using the link above. I did a very similar proposal five three years ago, but some work has been done on mediawiki2latex, so I propose it again.
Yours [[User:Dirk Hünniger|Dirk Hünniger]] ([[User talk:Dirk Hünniger|discuss]] • [[Special:Contributions/Dirk Hünniger|contribs]]) 14:39, 14 January 2024 (UTC)

:From the educational perspective the export feature of [https://en.wikiversity.org/w/index.php?title=Special:Book&amp;bookcmd=book_creator Wiki Book Creator] makes a lot of sense, because teachers can create a [[Risk Management/Tailored Wikibooks|tailored Wiki Book]] for each student, matching requirements and constraints of the learner. E.g. to support in more detail in specific topics and give more advanced learning activities in others. Technically the [https://en.wikiversity.org/w/index.php?title=Special:Book&amp;bookcmd=book_creator Wiki Book Creator] had already the feature years ago, so the [https://mediawiki2latex.wmflabs.org/ mediawiki2latex]  makes that usable again for the community. Maybe it make sense for other teachers as well. Thank you, [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 07:01, 19 March 2024 (UTC)
::Yes you are right Wiki Book Creator had these features before. But for many years Wiki Book Creator has disabled any possibilities to download books in any downloadable format. In Wiki Book Creator it is today only possible to order printed copies for a fee. In mediawiki2latex you can still download PDF, EPUB, LaTeX and Odt for free. [[User:Dirk Hünniger|Dirk Hünniger]] ([[User talk:Dirk Hünniger|discuss]] • [[Special:Contributions/Dirk Hünniger|contribs]]) 11:37, 19 March 2024 (UTC)

== Suggestion for Color Boxes ==

I am editing a course page, and I am trying to design it so that exercises are &quot;inside the text&quot; so to speak.  The intent is that the reader should never just be reading but always participating.  

But because of that structure, I feel like it's often visually unclear where the exercise ends and the exposition picks back up.  Therefore I'd like to put the exercises into some kind of a delimited box -- like a color box -- very similar to how definitions can be put into a color box with the &quot;Definition&quot; template.  However, as far as I can tell there are no templates for exercises.  

So I have two questions.

(1) How can I make color boxes?  I've googled around for this but most color boxes seem intended for single-line and inline uses, whereas most exercises are multi-line.  (I apologize if this is a dumb question, I'm not super handy with the technical aspects of Wikis.)

(2) Would it make sense to make an official exercise template like there is a template for definitions?  It seems like that would be a common enough thing that we might want it semi-standardized across Wikiversity.  Or perhaps specifically a template for math exercises, if there is a reason for those to be distinctly styled?  

Thanks for any help!  [[User:Addemf|Addemf]] ([[User talk:Addemf|discuss]] • [[Special:Contributions/Addemf|contribs]]) 18:38, 14 January 2024 (UTC)

:I suggest something like [[Template:inputcolorvariantexercise]], with he colors and the design one can play. I would also use the same template for showing exercises. If you then want to change the design, you can do it on the template. The following is the command when you want to include the exercise [[Functions/R/Strongly increasing/Injective/Exercise]]. It is best when the exercises are written somewhere else on neutral ground, so everybody can use them by inserting them with different styles.
&lt;nowiki&gt;
{{
inputcolorvariantexercise
|Functions/R/Strongly increasing/Injective/Exercise|m|
}}&lt;/nowiki&gt;
gives
{{
inputcolorvariantexercise
|Functions/R/Strongly increasing/Injective/Exercise||
}}
One can also do so that you can write the exercise text directly in your main text. It is also possible to make a variant with a solution (to expand, say). Many things are possible. But I would not strive for an offical how to present exercises, as people like different styles. Also note that the style of the exercise itself is different from the style presented by inserting the exercise. [[User:Bocardodarapti|Bocardodarapti]] ([[User talk:Bocardodarapti|discuss]] • [[Special:Contributions/Bocardodarapti|contribs]]) 18:55, 15 January 2024 (UTC)

{{robelbox|title=A reply|theme=2}}
The {{tl|robelbox}} template can be used to put content in colored boxes, like this reply is. However, be aware that it prevents the visual editor from being used normally on your page, and can make text harder to read; I'd recommend that you avoid using it for large stretches of content.
{{robelbox/close}}
[[User:Omphalographer|Omphalographer]] ([[User talk:Omphalographer|discuss]] • [[Special:Contributions/Omphalographer|contribs]]) 20:02, 14 January 2024 (UTC)

== the unprotection of making a blog post ==

&lt;span data-templatescript=&quot;WikiSign.js&quot; class=&quot;sign-button mw-ui-button mw-ui-progressive&quot;&gt;{{{1|Sign}}}&lt;/span&gt;
Hi folks, i noticed that making a blog post is protected. I understand why but i must request for it to be lifted cause young bloggers like me need the opportunity to get ratings on our work. [[User:Yellow Mellow Madie|Yellow Mellow Madie]] ([[User talk:Yellow Mellow Madie|discuss]] • [[Special:Contributions/Yellow Mellow Madie|contribs]]) 15:19, 15 January 2024 (UTC)

:How's that? Where are you trying to make a blog post? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 19:54, 15 January 2024 (UTC)

== Informing you about the Mental Health Resource Center and inviting any comments you may have ==

Hello all! I work in the Community Resilience and Sustainability team of the Wikimedia Foundation. The [[metawiki:Mental Health Resource Center|Mental Health Resource Center]] is a group of pages on Meta-wiki aimed at supporting the mental wellbeing of users in our community.

The Mental Health Resource Center launched in August 2023. The goal is to review the comments and suggestions to improve the Mental Health Resource Center each quarter. As there have not been many comments yet, I’d like to invite you to provide comments and resource suggestions as you are able to do so on the Mental Health Resource Center talk page. The hope is this resource expands over time to cover more languages and cultures. Thank you! Best, [[User:JKoerner (WMF)|JKoerner (WMF)]] ([[User talk:JKoerner (WMF)|discuss]] • [[Special:Contributions/JKoerner (WMF)|contribs]]) 21:33, 18 January 2024 (UTC)
: I think this needs community attention rather than custodian action, should this be moved to [[Wikiversity:Colloquium]]? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 07:41, 19 January 2024 (UTC)
::Yes. Doing it now. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:38, 19 January 2024 (UTC)
::: Thank you for the move. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 07:36, 31 January 2024 (UTC)

== Vote on the Charter for the Universal Code of Conduct Coordinating Committee ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/wiki/Universal Code of Conduct/Coordinating Committee/Charter/Announcement - voting opens|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:wiki/Universal Code of Conduct/Coordinating Committee/Charter/Announcement - voting opens}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''
Hello all,

I am reaching out to you today to announce that the voting period for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Universal Code of Conduct Coordinating Committee]] (U4C) Charter is now open. Community members may [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Charter/Voter_information|cast their vote and provide comments about the charter via SecurePoll]] now through '''2 February 2024'''. Those of you who voiced your opinions during the development of the [[foundation:Special:MyLanguage/Policy:Universal_Code_of_Conduct/Enforcement_guidelines|UCoC Enforcement Guidelines]] will find this process familiar.

The [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Charter|current version of the U4C Charter]] is on Meta-wiki with translations available. 

Read the charter, go vote and share this note with others in your community. I can confidently say the U4C Building Committee looks forward to your participation.

On behalf of the UCoC Project team,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 18:09, 19 January 2024 (UTC)
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== Test tools? ==
&lt;small&gt;&lt;s&gt;&lt;nowiki&gt;&lt;span data-templatescript=&quot;WikiSign.js&quot; class=&quot;sign-button mw-ui-button mw-ui-progressive&quot;&gt;{{{1|Sign}}}&lt;/span&gt;&lt;/nowiki&gt;&lt;/s&gt;&lt;/small&gt;

Does Wikiversity have test tools such as true/false questions, multiple choice, matching, sorting, quizzes, etc.? These are tools that are used in Moodle, for example. Where can I find them and how can I integrate them into Wikiversity? Thanks for help. [[User:Matutinho|Matutinho]] ([[User talk:Matutinho|discuss]] • [[Special:Contributions/Matutinho|contribs]]) 13:04, 26 January 2024 (UTC)
:A good place to start is '''[[Help:Quiz]]'''. There is a link at the top of that page to '''[[Help:Quiz-Simple]]''', which is a great entry point.  I spent a great deal of time making quizzes for physics and astronomy a few years ago.  My effort is at '''[[Quizbank]]'''.  I eventually migrated the project to '''[https://www.myopenmath.com/ www.myopenmath.com]'''.  If you are going to seriously do quizzes, myopenmath.com is superior.  I finally settled in on a method whereby the quizzes are on Myopenmath, but the hints and ancillary materials were on Wikiversity's [[Quizbank]] (or another Wikiversity page.)  To this day I get enough pageviews on some of my [[Quizbank]] pages to know that my questions are being used.  I don't know how and by whom.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:25, 26 January 2024 (UTC)

== Last days to vote on the Charter for the Universal Code of Conduct Coordinating Committee ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/wiki/Universal Code of Conduct/Coordinating Committee/Charter/Announcement - voting reminder|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:wiki/Universal Code of Conduct/Coordinating Committee/Charter/Announcement - voting reminder}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''
Hello all,

I am reaching out to you today to remind you that the voting period for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Universal Code of Conduct Coordinating Committee]] (U4C) charter will close on '''2 February 2024'''. Community members may [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Charter/Voter_information|cast their vote and provide comments about the charter via SecurePoll]]. Those of you who voiced your opinions during the development of the [[foundation:Special:MyLanguage/Policy:Universal_Code_of_Conduct/Enforcement_guidelines|UCoC Enforcement Guidelines]] will find this process familiar.

The [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Charter|current version of the U4C charter]] is on Meta-wiki with translations available. 

Read the charter, go vote and share this note with others in your community. I can confidently say the U4C Building Committee looks forward to your participation.

On behalf of the UCoC Project team,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 17:01, 31 January 2024 (UTC)
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== Wikidata for Beginners ==

Hey, I am hosting a [[User:Juandev/LDW24/wd|Wikidata for Beginners workshop]] on Wednesday, 14 February 2024, as a part of Love Data Week 2024 (LDW), so you are welcome to attend. I would like to ask you to keep an eye on that landing page to prevent any vandalism, as it's linked to the [https://www.icpsr.umich.edu/web/ICPSR/cms/5224 LDW page], too. Thx. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:19, 11 February 2024 (UTC)
: Semi-protection {{done}} by [[special:redirect/logid/3387071]]. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:48, 11 February 2024 (UTC)

== Announcing the results of the UCoC Coordinating Committee Charter ratification vote ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/wiki/Universal Code of Conduct/Coordinating Committee/Charter/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:wiki/Universal Code of Conduct/Coordinating Committee/Charter/Announcement - results}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Dear all,

Thank you everyone for following the progress of the Universal Code of Conduct. I am writing to you today to announce the outcome of the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Charter/Voter_information|ratification vote]] on the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Charter|Universal Code of Conduct Coordinating Committee Charter]]. 1746 contributors voted in this ratification vote with 1249 voters supporting the Charter and 420 voters not. The ratification vote process allowed for voters to provide comments about the Charter. 

A report of voting statistics and a summary of voter comments will be published on Meta-wiki in the coming weeks.

Please look forward to hearing about the next steps soon.

On behalf of the UCoC Project team,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 18:24, 12 February 2024 (UTC)
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==How to handle very-low-value pages AKA deletion and move to userspace convention==
I am starting this discussion based on a prelude:
* [[Wikiversity:Requests for Deletion#Wikiversity:Deletion Convention 2024]]
I and Guy vandegrift differ at times about what belongs to mainspace. Guy has been doing a lot of tireless deletion/move-to-userspace work, often based on my proposals; thank you! Some of the deletion proposals resulted in RFD discussions at [[Wikiversity:Requests for Deletion]]. An example of where we see things differently is [[Student Projects/PhotoTalks]], which I find not good enough for mainspace. The relevant guideline (not policy) is [[Wikiversity:Deletions]]; the key phrase is &quot;learning outcomes are scarce&quot;.

I will let Guy pick the questions he wants to put forward for discussion. Reposts from the linked discussion are perhaps not amiss.
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:22, 16 February 2024 (UTC)

Let me violate the above a bit, and put one item for me into the discussion. As a point of contrast: I find the page [[Historical Introduction to Philosophy/Truth, Objectivity, and Relativism]] to be of rather low quality: there are too many dubious statements and there is a conspicuous lack of good further reading specific to the subject of the subpage. But it is not the kind of page that I would send for deletion as part of the current cleanup effort. The kind of page that I am sending for deletion is [[Student Projects/PhotoTalks]], which is not a &quot;project&quot; in a meaningful sense and from which the reader can hardly learn anything. And I have no qualms with &quot;PhotoTalks&quot; being moved to user space, although I find it too kind anyway; but I have no fundamental problem with this kind of arguably great-than-expected kindness. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:31, 16 February 2024 (UTC)

* {{Comment}} I was pinged because I made some comments etc. about deleting media files. Mostly those deletions were suggested because of copyright issues.
: I understand that Wikiversity is a place to learn and study and in all levels. And sometimes you can learn by other peoples mistakes. But it raises 2 problems:
:# If a page is of low quality and it contains mistakes should it at least be flagged as low quality and with mistakes? Otherwise someone may learn something wrong. But who check all the pages and make sure the quality is okay? An when should this happen? If I make a page for a school project then it would not give a fair impression of my skills if other users starts to correct my errors. So it should not happen untill after the project is over and my skills have been evaluated. 
:# Just because we can learn from eachothers mistakes does that mean we should keep everything? On Commons [[:c:Commons:Project_scope#File_not_legitimately_in_use|Scope page]] it says: &quot;For example, the fact that an unused blurred photograph could theoretically be used to illustrate an article on &quot;Common mistakes in photography&quot; does not mean that we should keep all blurred photographs.&quot; I think the same could apply here.
:Anyway I think it is a very good idea to agree on some guide/policy etc. because I think it will make it much easier for everyone. But I do not think I can contribute very much to that because I'm not really active outside the file namespace. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 08:47, 16 February 2024 (UTC)

*My answer to all this can be found on '''[[Wikiversity:Deletion Convention 2024]]'''--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:21, 16 February 2024 (UTC)
*: There is a lot to respond to in [[Wikiversity:Deletion Convention 2024]]. I will make it brief, to save the attention of everyone, but I can post more if wished and answer any questions anyone has for me.
*: 1) The quote &quot;Too many bad articles and we don't have the time to remove them, too few bad articles, and there is no need delete them&quot; provides a recipe to keep a growing number of very-low-value pages in the mainspace, which cannot be a good thing.
*: 2) Very-low-value pages should IMHO ideally either be moved to user space or deleted; they should not stay in the mainspace.
*: 3) Page &quot;[[Finding and using free content]]&quot; should be deleted; two of the three links do not show valuable content (are quasi-broken) and the 3rd link is an internal one.
*: 4) No admin should feel compelled to do most of the deletion work alone. One option is to do the deletion work only on, say, Tuesdays and only delete, say, at most 7 pages per Tuesday, to give other admins plenty of time to join the effort.
*: --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:51, 17 February 2024 (UTC)

*{{Ping|Dan Polansky}} I woke up this morning with an idea that is directly related to your point 1) directly above. I believe that idea will render the other points (2-4) moot: '''Simply move the page to draftspace and leave a redirect.''' Also, I '''strongly oppose''' not leaving a redirect in case the student wants to come back and read or edit the page. (rewritten)-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:43, 17 February 2024 (UTC)
*: The problem with the Draft namespace is this (per [[Wikiversity:Drafts]]): &quot;Resources which remain in the draft space for over 180 days (6 months) without being substantially edited may be deleted.&quot; And thus, the deletion is only deferred anyway (or does &quot;may&quot; mean deletion is just an option taken on a whim?), but not very much, but the process then takes more work/more steps.
*: Students can easily find their pages in their contribution list (e.g. [[Special:Contributions/Dan Polansky]]), which should be easy to overview unless the student was very prolific. Therefore, keeping redirects seems inessential. And if the moving is to userspace, finding the contribution is also easy, using a template that lists userspace subpages. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:04, 19 February 2024 (UTC)
:::Three items:
:::#What about moving to userspace with a redirect, and with template at the top of the page identifying that page as being in the person's userspace?
:::#Also, I just got a thank you from a person whose article I moved to userspace instead of deleting.
:::#No consensus is being formed here, and if nothing happens I will no choice but to use my authority as a Custodian and impose something.  What I do will be based largely on previous practice, because as you know, our stated policy guidelines were never taken seriously.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:19, 19 February 2024 (UTC)
* Since I'm really not active here I would hate to make things more complicated. But I just saw [[Wikiversity:Requests_for_Deletion#Student_Projects%2FPhotoTalks]] and in my opinion this should be deleted as a speedy per [[Wikiversity:Deletions#Criteria]] #1. I see no reason to even discuss it.
: If it was a new page and the user was still working on it then I would not judge it too hard. But it has not been edited for many years.
: According to [[Wikiversity:What is Wikiversity?]] Wikiversity is a learning community for learning, teaching, researching, serving and sharing materials and ideas. Can anyone explain why the page meets any of that? Who will learn anything from the page. What would they learn from it? Is there any research in this? Etc. 
: I do not agree that we gain (almost) nothing by deleting low quality pages. If anyone searches then junk will also show up. If users see too much junk it will give the impression that this project is a low quality project. Personally I would not use a project if I know that there are no minimum/quality requirements.
: I fully understand that it is a huge task to clean up and I know there are cases where someone might disagree. But it should be possible for those that disagree to provide some good arguments why the page should not be deleted. So perhaps modify {{tl|Prod}} a bit so that if anyone wants to remove the template they should at least add a reason on the talk page.
: So my suggestion is delete vandalism etc. at once. If there could be any doubt add {{tl|Prod}}, wait 30 days (or whatever) then delete. If someone disagree they should be required to provice a realistic argument why it is not a deletion. If nominator does still not agree the page should be kept then start a formal deletion and hope there is anyone else that would like to comment. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 17:35, 19 February 2024 (UTC)

::The following comment preceded the previous one, but was placed under the wrong section by the author Guy vandegrift: '''Why are we talking about [[Student Projects/Geography]]???''' It's a student project and nobody is going there to look for ideas about teaching. Pageviews count the times an editor looks at the page, but if you look at the pageviews after the page was completed [https://pageviews.wmcloud.org/?project=en.wikiversity.org&amp;platform=all-access&amp;agent=user&amp;redirects=0&amp;start=2021-04&amp;end=2024-01&amp;pages=Student_Projects/Geography '''this is what you get.''']. Also look at '''[[Student_Projects#Student_Pages]]'''. I think it's almost 300 pages.  In the experimental sciences we learn to make estimates.  I estimate that it will take 100 person hours to get the references right on all these pages. I don't want to waste one more hour of my time on this.  In 2020 a Bureaucrat Dave Braunschweig allowed the page to be at its current location. Why am I being asked to revert that decision?  I repeat: Nobody cares about ''Student_Projects/Geography''. It does no harm, except that talking about it wastes time and space on the Colloquium.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:05, 19 February 2024 (UTC)
::: For clarification, I did not nominate [[User:PURNA BISWAS2/Student Projects/Geography]] for deletion (when it was at [[Student Projects/Geography]]), although it indeed does not belong to mainspace, IMHO. I am focusing on ''top-level'' mainspace pages with arguably unacceptable quality/implemented scope. To give an idea for what I mean, I just used &quot;Random page&quot; wiki function to find the following pages arguably worthy of deletion/moving to userspace: [[The Distribution of Addition and Subtraction over Multiplication in Elementary Algebra]], [[The iam conjecture]], [[Web Design:Useful Books]], [[Internet Abuse]], and [[Wikitext 101]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:15, 22 February 2024 (UTC)

::At the moment, the &quot;vote&quot; is 2 for deletion, with only me wanting to keep the page. It looks like Wikiversity is now refereeing the quality of student efforts.  As per the old (informal) policy, I will move it to the author's userspace. --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:42, 19 February 2024 (UTC)

::: How do you know that [[Student_Projects/PhotoTalks]] was created as a Student Project? But even if it was there is no rule saying that it has to be kept online forever? Student Projects could be deleted after some time - one year for example. As you said nobody is going there to look. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 06:53, 20 February 2024 (UTC)
: I support deleting as little as possible that is not blatant spam. I think deleting user creations discourages users from actively using the wiki. Deleting any content that is not blatant spam that may have been created in good faith may actually be a form of punishment effectively (from a behavioral psychology point of view). If something is not actual spam, then IMO it should not ever be deleted. It should either be moved to draft namespace or user namespace... or like a &quot;Recycle bin&quot; so others could access and reuse/repurpose the content or utilize it later - even if it is just a bare bones minimal page (like a stub). Deleting content I think will continue to hinder this wiki from growing and reaching its potential. ChatGPT reached 100 million users in a few months? And this wiki has existed for how long and has how many users? This wiki has so much potential but I often stop myself from editing here and tell myself it is not a good use of time because something I may create in good faith may be deleted (thereby wasting my time and efforts). Limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 15:55, 22 February 2024 (UTC)
:: The key question of this thread is not when things should be ''outright deleted'' but rather when can they be ''moved out of mainspace''. And if I read the above correctly, it opposes the former (deletion) but not the latter (move to userspace); correct me if I am wrong. (A wiki is nothing like a chatbot; not much point comparing the two.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:03, 22 February 2024 (UTC)
:: In section [[#Expanding WV:Deletions with provision for moving to user space]], I proposed to codify moving to userspace as common. Even stronger language could be used than I used, in favor of moving to user space. This could lead people to think that even if their creation gets removed from mainspace, it will at least end up in their user space. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:05, 22 February 2024 (UTC)
:::Then I shall clarify. If pages are kept in main namespace, OK. If they are moved to userspace or draft namespace then those pages should be organized and linked to in such a way that they are not effectively impossible or extremely difficult to find or notice (if one's intention is to see less developed content and stubs for possible development). I oppose outright deletion for reasons I noted. Bless up. 06:17, 24 February 2024 (UTC)

==What to do about [[WV:Verifiability]]==
My edits to the page have been challenged elsewhere, but a proper venue seems to be Colloquium. We should figure out what to do about them.

1) Revert to the state before my edits. I find this suboptimal since I find these edits to be an improvement, but I am no dictator here.

2) Keep in the state in which I left the page. Still far from ideal, but at least some defects have been addressed.

3) Make amends to the state in which I left the page.

I will point out that even after the changes I made, the page is at odds with the actual widespread practice. One only has to look e.g. at [[Student Projects]] and its subpages to see that requirements of either reference-verifiable statements or original &quot;scholarly research&quot; are being largely ignored. To wit, e.g. [[Student Projects/Geography]] contains only one external link (to youtube) and surely is not &quot;original scholarly research&quot; by any standard; it is a rather unoriginal yet original in the sense of copyright law writeup, perhaps by a student. [[Student Projects]] has other such pages.

Ideally, we would figure out how to amend [[WV:Verifiability]] to match the intended tolerance for unreferenced texts in Wikiversity. In the meantime, it seems advisable not to pretend the page is a binding policy that is actually enforced. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:09, 19 February 2024 (UTC)

:I am neutral on this.  If you follow the rules, we should revert your changes because no census was reached. If we follow past policy, no action needs to be taken because much of what was written was never properly voted on.  Wikiversity is a very small organization. We are so small in number that we can either improve the wiki or improve the rules, but not both, IMHO.  I am getting caught up in all this because I am one of the few Custodians who is deleting pages, and I might stop doing that (I'm only a volunteer.) ... Also, it's OK to discuss things on the Colloquium, but decisions need to be done in Wikiversity space because: (1) That's the way we used to do it, and (2) these decisions take a long time (many months) and discussions get archived or lost before everybody votes.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 20:39, 19 February 2024 (UTC)

==Expanding WV:Deletions with provision for moving to user space==
I propose to add something like the following to [[WV:Deletions]]:
:&lt;nowiki&gt;&quot;==Moving to user space==&lt;/nowiki&gt;
:&quot;A page that meets the criteria for deletion can be moved to user space instead, unless an overriding rationale for deletion prevails such as the page being offensive, copyright violation, etc. Rationale: The database storage is not saved by deletion and there is generally no harm in being kind to those who hone their writing and wiki editing skills in Wikiversity.&quot;
Thoughts? Any supports? Phrasing modification proposals? Should the rationale be omitted? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:39, 20 February 2024 (UTC)
::The policy you proposed is better than what we have.  But I keep coming back to the &quot;calculation&quot; of what is and is not possible. We need to move fast and efficiently because removing less than 10% of the low quality pages accomplish nothing. It is time consuming to go into the history and decide who the author(s) were.  I propose [[Draft:Archive/Pagename]] for all such pages.  This will allow people to search [[Draft:Archive]] to locate their work.  I also propose that we give high priority to two distinctly different types of pages:
::#Old pages that have been dormant for 5 years or more.
::#Vast quantities of new pages that a hyperactive newbie creates.  Half of them are doing real harm with nonsense pages, and need to be asked to leave (or at least work in userspace.) The other half need to be encouraged to work under one or two subpages.
::--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:51, 22 February 2024 (UTC)
::: Are the pages in &quot;Draft:Archives/...&quot; exempt from &quot;Resources which remain in the draft space for over 180 days (6 months) without being substantially edited may be deleted&quot;, which was voted on in [[Wikiversity talk:Drafts#Draft namespace resource retention]] in April 2019? If they are not exempt, why should a page creator prefer the Draft page over user space, in which the material can be left alone indefinitely? I for one would prefer my writings to end up in my user space and stay there &quot;forever&quot;, publicly accessible.
::: Yes, you are right that figuring out the right user is more work and sometimes may be harder to do or impossible. For that scenario, the draft space seems to be a fine option.
::: Shouldn't we codify both options, as &quot;can&quot;? We would start by adding the user space option as proposed above, and I would propose another option for the Draft space in a separate thread? (Or you could do it if preferred.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 19:33, 22 February 2024 (UTC)
::::Yes, the Draft:Archive would be exempt from removal. Putting the material in Draft:Archive has lots of advantages: (1) Virtually all pages have multiple authors (2) people can easily search this DraftArchive space using either Google or Wikiversity's search option. (3) Nobody in their right mind is going to judge Wikiversity by what they read in a space called &quot;Draft:Archive&quot;.  What distinguishes Wikiversity from the other wikis is the we [[Wikiversity:Learning by doing|&quot;learn by doing&quot;]], and we all learn from our mistakes. Before we propose this option to the community, I suggest we just do it for about ten pages, and see who complains. 

::::Also, the page created used the singular: See '''[[Draft:Archive]]'''. Feel free to replace ''Lorem ipsum'' with proposed guidlines.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 20:13, 22 February 2024 (UTC)
::::: Since the proposed plan is to let very-low-quality material sit somewhere in the Draft:Archive/... indefinitely, I think the better plan is to let sit all pages in the Draft:... space indefinitely, which would require a formal abolishment of [[Wikiversity talk:Drafts#Draft namespace resource retention]] via new voting somewhere. But even if we want to have  Draft:Archive/... exempt from expiry, it probably requires a process as formal as the linked vote, doesn't it? I struggle to find the significant difference between the kind of material that belongs to Draft/... and the kind that belongs to Draft/Archive/... --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 20:26, 22 February 2024 (UTC)
::::::There is a way out of any legal conundrums associated with [[Draft:Archive]]: Let's consider it my personal project, always under construction.  That way I don't need anyone's permission to maintain the page, unless someone can make the argument that it hurts Wikiversity. I haven't read the Wikiversity policy, but I doubt there is a deletion date for drafts that are still being edited. I have certain rules for this project. The following items are not allowed, for example:
::::::#Bad attempts at humor, or commercial advertising (this eliminates most spam)
::::::#Excessive pages by a single editor
::::::#All hate speech, and any pseudoscience that is patently false
::::::* Also, any link, image, or template that interferes with Wikiversity can be &quot;dewikified&quot; using &lt;code&gt;&lt;nowiki&gt;&lt;nowiki&gt;...&lt;/nowiki&gt;&lt;/nowiki&gt;&lt;/code&gt;
::::::As I was looking for pages to place in this archive, I took a second look at [[Student Projects/PhotoTalks]]. Those who want to remove it from its subspace may outnumber me. But they are wrong. I spent several years at a University consulting with primary and secondary teachers on the teaching of math and science (my efforts were largely useless because at that level teaching is 99% babysitting and 1% content.) But assigning a young student to learn how to create a page on Wikiversity with images is an excellent thing to do.  Even if the person who created the page was not a child.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 23:33, 22 February 2024 (UTC)
::::::: Returning to the original topic, do you have any objections to expanding the [[WV:Deletions]] page as proposed, to codify moving to user pages? The text says &quot;can&quot; (an option), captures actual recent practice and does not preclude using Draft namespace for a similar purpose. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 23 February 2024 (UTC)
:::As I am a newbie and have been somewhat active, I wonder whether I am doing things the right way.  
:::I don't want to bother anyone by asking them to review what is quite a lot of writing. But if I have been doing anything which has stood out as less than ideal, please feel free to let me know.   [[User:Addemf|Addemf]] ([[User talk:Addemf|discuss]] • [[Special:Contributions/Addemf|contribs]]) 19:41, 23 February 2024 (UTC)
::::{{ping|Addemf}} Everything you wrote looks OK to me.  Your topic is outside my field of expertise, though. If you need any help, click &quot;discuss&quot; after my username:[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:55, 24 February 2024 (UTC)
:::::Thanks! As long as none of the style or ways that I'm editing are upsetting to anyone, I'm pleased!  :) [[User:Addemf|Addemf]] ([[User talk:Addemf|discuss]] • [[Special:Contributions/Addemf|contribs]]) 17:56, 24 February 2024 (UTC)

==Delete pure junk==
I think pure junk should be deleted rather than moved to userspace or draftspace.

For instance: [[Draft:Istanamshjs]]. The title is nonsense, the &quot;content&quot; is nonsense; not even sentences.

This kind of &quot;contribution&quot; is not worth anyone's time, not even the time to push the &quot;move&quot; button. The only button this deserves is &quot;delete with extreme prejudice&quot;.

[[WV:Deletions]] does not disagree, or outright agreees.

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:29, 24 February 2024 (UTC)

:{{ping|Dan Polansky}} You are asking to delete this page approximately one day after it was created.  It is perfectly acceptable to start a draft with an outline.  There is a bit of justifiable prejudice against IP editors because they tend not to stay long. But perhaps the person is being cautious about registering with an unfamiliar organization.  I still haven't made the decision to sign up for CNN. I admit that the draft looks incoherent, and give this project a 5% chance of going anywhere, and that probability will go down every day this draft is not edited.  This leaves me with three questions: (1) What probability of success is the threshold for deleting a draft? (2) How do we obtain that probability? By one person's opinion, or do we need some sort of consensus? (3) At what point do we reach a point of diminishing return on our efforts? In other words, bringing this up on the Colloquium or RFD wastes our time. Just give it a &lt;code&gt;&lt;nowiki&gt;{{subst:prod}}&lt;/nowiki&gt;&lt;/code&gt; and move on.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:40, 24 February 2024 (UTC)
::{{Ping|Guy vandegrift}} {{tl|ping}} only works if you also sign a post. So if you go back and edit a comment to add a ping, you should also re-sign that post. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:07, 24 February 2024 (UTC)
:::{{ping|Koavf}} I'm confused: I thought I did sign the post.  Perhaps my numbered list confused people. I will re-edit, but not resign because that confuses the record.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:29, 24 February 2024 (UTC)
::I submit that [[Draft:Istanamshjs]] is not a draft of anything but a pure junk. To my mind, the page is not acceptable at all, and I don't care about whether the author was an IP in this instance.
::I brought this to Colloquium so that we can agree that some pages are junk enough to be deleted outright rather than being &quot;salvaged&quot; in Draft space. If there is no such agreement, we can as well rename &quot;Draft&quot; space to &quot;Dustbin&quot;.
::I find this lenience toward pure junk very perplexing. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:39, 25 February 2024 (UTC)
::For the record, I quote this junk in full:
&lt;nowiki&gt;&lt;small&gt;&lt;/nowiki&gt;
&lt;pre&gt;
This page Istanamshjs Does not exist

This Page but,Always you can Make a new draft for                           

Review.


Did you mean Islam JTK?

Islam 1988 Monopoly

Islam 1991 User

Islam 1996 

Islam 1998

Islam 1999

Islam 2000 Event help

Islam 2021

Privacy Policy  Terms  Help of Service   Feedback  Advice  Settings 


Sign in  Sign Up
&lt;/pre&gt;
&lt;nowiki&gt;&lt;/small&gt;&lt;/nowiki&gt;
This is not a draft. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:40, 25 February 2024 (UTC)

== [[Wikiversity:Requests_for_Deletion]] ==

I am planning to close a number of discussions at '''[[Wikiversity:Requests_for_Deletion]]'''. A few of them have unresolved issues, but the discussions are so contorted that we need to resolve those issues with fresh starts for each issue. I will collapse the discussions to leave room for you to list any items that need to be resolved.
#[[Wikiversity:Requests_for_Deletion#Wikiversity:Deletion_Convention_2024]]
#[[Wikiversity:Requests_for_Deletion#Thousands_of_unused_files]]
#[[Wikiversity:Requests_for_Deletion#Invalid_fair_use_by_User:Marshallsumter]]
#[[Wikiversity:Requests_for_Deletion#Draft:Proof_for_NP_unequal_P_by_Thomas_Käfer]]
#[[Wikiversity:Requests_for_Deletion#PhotoTalks]]
#[[Wikiversity:Requests_for_Deletion#Wikisphere]]
#[[Wikiversity:Requests_for_Deletion#Ukulele]]
Yours truly, [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 04:51, 25 February 2024 (UTC)
: Resolvable outcomes based on existing discussion and [[WV:Deletions]] guideline:
:[[Wikiversity:Requests_for_Deletion#Wikiversity:Deletion_Convention_2024]]: not a specific deletion proposal =&gt; no action taken.
:[[Wikiversity:Requests_for_Deletion#Thousands_of_unused_files]]: not clear
:[[Wikiversity:Requests_for_Deletion#Invalid_fair_use_by_User:Marshallsumter]]: not clear; something was already done
:[[Wikiversity:Requests_for_Deletion#Draft:Proof_for_NP_unequal_P_by_Thomas_Käfer]]: delete.
:[[Wikiversity:Requests_for_Deletion#PhotoTalks]]: move out of mainspace (2:1)
:[[Wikiversity:Requests_for_Deletion#Wikisphere]]: move out of mainspace (already done, but redirect was kept)
:[[Wikiversity:Requests_for_Deletion#Ukulele]]: delete (or move out of mainspace).
:On a process/administration note, I do not find collapsing discussions useful at all, not even before archiving. The English Wiktionary does not collapse its RFD discussions. The English Wiktionary closes each discussion with a closure statement, e.g. &quot;RFD-kept&quot; or &quot;RFD-deleted&quot;. It would be great to proceed in a similar fashion. From the archives of [[Wikiversity:Requests_for_Deletion]], I see the English Wikiversity did not use to ''collapse'' discussions. For instance, [[Wikiversity:Requests for Deletion/Archives/16]] uses boxed closures using {{tlx|archive top}} and {{tlx|archive bottom}} but no collapsed closures; and each boxed closure states the specific outcome. I think even these boxes with blue background are unnecessary and make the archived discussions harder to read. The phrasing of the closing statement is not that important; it can be &quot;Deleted per consensus&quot;, &quot;No consensus for deletion&quot;. One can emphasize closing if one wishes: &quot;Closed: deleted per consensus&quot;, &quot;Closed: no consensus for deletion&quot;, etc.
: If discussions are to be collapsed, the collapsing should be on a per discussion level rather than multiple discussions being in a single collapsed section (perhaps that was just an editing error?)
: On a time note: closing Ukulele is too early IMHO since not even one week elapsed. We should agree on a minimum period for keeping RFD discussions open and keep that period. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:35, 25 February 2024 (UTC)
::The confusion about [[Ukulele]] is my fault. I closed the discussion (&quot;turned it blue&quot;) and collapsed it because there was a consensus to delete. I will not move the discussion into the archive (&quot;hide it&quot;) until it is deleted.  I need to prioritise my efforts, and deleting well designed stubs is not a high priority.  I said this is my fault because I forgot to change the tag on Ukalele from &quot;rfd&quot; to &quot;speedy&quot;.  Now anybody who wants to delete will see it on [[:Category:Candidates_for_speedy_deletion]].[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:34, 28 February 2024 (UTC)
::: I think that [[Wikiversity:Requests_for_Deletion#Invalid_fair_use_by_User:Marshallsumter]] Can be closed. Someone just have to decide if the pages made by MS and/or Kizer should be deleted or not. If someone know the answer to the off topic question it would be Great but if not that page could be discussed in a new RFD.--[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 05:55, 29 February 2024 (UTC)
:::: The start of [[Wikiversity:Requests_for_Deletion#Invalid_fair_use_by_User:Marshallsumter]] mentions &quot;out invalid fair use files&quot;; are there still any ''files'' (not pages) by User:Marshallsumter that someone considers problematic? Either way, it is probably a good idea to close this old RFD thread, and open a new one if required, say, &quot;Files by User:Marshallsumter with invalid fair use&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:27, 29 February 2024 (UTC)
:::::I created a space for this discussion at '''[[Wikiversity:Requests_for_Deletion#Archiving_of_Invalid_fair_use_by_User:Marshallsumter]]''' --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 08:03, 29 February 2024 (UTC)
:::::: Good. The title is a bit ambiguous: are we talking ''files'' or ''pages''? Anyway, let us see what discussion develops there. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:15, 29 February 2024 (UTC)
::::::: Invalid fair use files are a copyright problem. My plan was to delete pages in user namespace. Once the pages were deleted many files would be orphan and easy to find and delete. Very simple plan - at least in my head :-D However the files have been found and deleted now. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 16:23, 29 February 2024 (UTC)
::::::::I have no problem deleting user namespace pages that are causing trouble.  In fact, I'm still a bit concerned about moving &quot;dead&quot; pages to either userspace or [[Draft:Archive]] for exactly that reason.  If anybody ever runs into a problem caused by such pages, let me know.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:42, 29 February 2024 (UTC)

==Collapsing discussions in Requests for Deletion (RFD)==
I register my opposition to the use of collapsing in [[Wikiversity:Requests for Deletion]]. Collapsing the discussions makes it harder to skim the RFD page just by scrolling down and see what is going on there. As a weak argument, collapsing RFD discussions is not common in the English Wiktionary. One does not need collapsing to move from one discussion to another: there is a table of contents for that. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:51, 28 February 2024 (UTC)

== Should we allow custodians to use mass-delete? ==

At [[special:permalink/2609683#Does_anybody_know_how_to_delete_all_pages_by_a_single_user?]], there was a related discussion about this matter. I'm bringing this agenda here for the community's attention. As can be seen at [[Special:ListGroupRights]], only bureaucrats are allowed to use mass-delete under current settings, but many Wikimedia projects allow this to admins (equal to our custodians). Global sysops can also use mass-delete. What does our community think about this? Should we keep the current settings, or should we grant mass-delete to our custodians as a new standard? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:40, 1 March 2024 (UTC)
: @[[User:MathXplore|MathXplore]] Sounds like a good idea to me - I can see value in [[Special:Nuke]] being available to custodians on en.wv. Consider starting a proposal at [[Wikiversity talk:Custodianship]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:24, 8 March 2024 (UTC)
:: I have started a proposal at [[Wikiversity_talk:Custodianship#Proposal_to_allow_custodians_to_use_mass-delete]] per suggestion. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 07:13, 8 March 2024 (UTC)

== Proposal for Integrating University Knowledge into Wikipedia ==
Hello! I share with you this proposal, it is still not very mature, I would appreciate ideas and suggestions to carry it out. Any opinion is welcome. Thanks

The aim of this proposal is to compile and share all academic content from universities on Wikipedia. This involves adding the syllabi of all subjects from all university degrees to the platform, providing free and open access to university knowledge for anyone or any artificial intelligence.

Implementation:

1. '''Acquisition and Digitalization of Notes:'''

*Students are encouraged to digitize their notes and publish them under the CC BY SA license for inclusion in Wikipedia.
*Participation from students can be promoted through annual awards for the best notes in each degree, whether in the form of monetary incentives, meal vouchers, or transportation grants. It is crucial to avoid plagiarism and respect copyright.
*Collaboration with student delegations and the involvement of professors in this task are encouraged.

2. '''Uploading Content to Wikimedia Commons:'''
*Students can upload their notes to the Wikimedia Commons repository with their full name, which enhances their visibility on Google and improves their curriculum vitae.

3. '''Distribution of Content on Wikipedia:'''
*The notes are reviewed and understood, and the information is distributed across related Wikipedia articles.
*To achieve this, two options can be considered:
#Hiring specialized editors.
#Requesting the collaboration of Wikipedia volunteers, possibly establishing a Wikiproject dedicated to organizing tasks and coordinating content contributions.

This proposal aims to enrich Wikipedia's content with verified and accessible academic knowledge, benefiting students, researchers, and learning enthusiasts worldwide. 

In commons there are already uploaded notes for several subjects, see [https://commons.wikimedia.org/wiki/Category:Lecture_notes Lecture Notes]
[[User:Uni4all|Uni4all]] ([[User talk:Uni4all|discuss]] • [[Special:Contributions/Uni4all|contribs]]) 17:58, 2 March 2024 (UTC)

:My advice is to start small. Go on Wikiversity and start a small project. Focus on quality versus volume. We are a community that shares your goals, but we live in a much larger and more diverse world that is not ready for your plan.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:35, 2 March 2024 (UTC)
: IMO to my knowledge, lecture notes could simply be added to this wiki. If there is a better way to think about that please LMK. Bless up! [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 02:21, 7 March 2024 (UTC)

==  Report of the U4C Charter ratification and U4C Call for Candidates now available ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024/Announcement – call for candidates| You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024/Announcement – call for candidates}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Hello all,

I am writing to you today with two important pieces of information. First, the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Charter/Vote results|report of the comments from the Universal Code of Conduct Coordinating Committee (U4C) Charter ratification]] is now available. Secondly, the call for candidates for the U4C is open now through April 1, 2024.

The [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Universal Code of Conduct Coordinating Committee]] (U4C) is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community members are invited to submit their applications for the U4C. For more information and the responsibilities of the U4C, please [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Charter|review the U4C Charter]].

Per the charter, there are 16 seats on the U4C: eight community-at-large seats and eight regional seats to ensure the U4C represents the diversity of the movement.

Read more and submit your application on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024|Meta-wiki]].

On behalf of the UCoC project team,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 16:25, 5 March 2024 (UTC)
&lt;!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=26276337 --&gt;

== Wikimedia Canada survey ==

Hi! Wikimedia Canada invites contributors living in Canada to take part in our 2024 Community Survey. The survey takes approximately five minutes to complete and closes on March 31, 2024. It is available in both French and English. To learn more, please visit the [[wmca:Form2024| survey project page]] on Meta. [[User:Chelsea Chiovelli (WMCA)|Chelsea Chiovelli (WMCA)]] ([[User talk:Chelsea Chiovelli (WMCA)|discuss]] • [[Special:Contributions/Chelsea Chiovelli (WMCA)|contribs]]) 00:19, 7 March 2024 (UTC)

== Archive namespace? ==

What about an Archive namespace? ... Its not exactly for drafts... but this could be utilized to sort of wall off Creative Commons content created in good faith from the main namespace that might otherwise be deleted? Or maybe Draft can serve this function, if such a function is desired? bless up. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 02:20, 7 March 2024 (UTC)
:Perhaps you already know about an informal version of this I started a few weeks ago at [[Draft:Archive]].--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:48, 7 March 2024 (UTC)
::I did not know that until today. That is awesome!! thanks and limitless peace! now I know! thanks. Bless up. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 08:21, 9 March 2024 (UTC)
:::{{ping|Michael Ten}} I hate to bother you about this, but now that you know about Draft:Archive, I presume you are willing to change your vote so that [[WikiService]] and be moved to [[Draft:Archive/2024/WikiService]]. You can change your vote at [[Wikiversity:Requests_for_Deletion#Voting_on_WikiService]] simply by deleting and rewriting.  Equivalently, you can affirm right here that we should move [[WikiService]] to [[Draft:Archive/2024/WikiService]], and I will report that affirmation and close the discussion over there. The page has had no significant edits since 2008, and the significant authors have been dormant for a decade (BTW I am OK with deleting something this old.)--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:57, 10 March 2024 (UTC)
:::: ''Age'' and ''dormant'' are not deletion criteria per [[WV:Deletions]]. A valuable page can be dormant and that is fine per [[WV:Deletions]], unless I have overlooked something. The problem with the page is that it is worthless and will not help readers learn anything; if the page were excellent since 2008 and without further changes, it would be kept. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:40, 10 March 2024 (UTC)
::::: &quot;The problem with the page is that it is worthless and will not help readers learn anything [...]&quot; I ultimately consider this a subjective value judgement if content (on this wiki that relates to learning, teaching, and/or research) was likely created in good faith and is not spam. Hence I think the proposal modification i noted here ([[Wikiversity:Colloquium#Expanding_WV%3ADeletions_with_Moving_to_Draft_archive]]) may be fruitful. I think [https://usdictionary.com/idioms/one-mans-trash-is-another-mans-treasure/ &quot;One man's trash is another man's treasure&quot;] quite much applies to Creative Commons educational content created in good faith. Suggested Google search which might have some relevant content (as food for thought) in the 3.8 million results I see per Google is as follows, &quot;subjective value educational content site:.edu&quot; (without quotes). Regardless, I appreciate your constructive good faith contributions and perspectives! bless up! [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 17:41, 10 March 2024 (UTC)

== Template:Draftify ==

I created [[Template:Draftify]] today. Does a template like this already exist? I applied this template to [[Metadata]]. bless up. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 08:22, 9 March 2024 (UTC)
: I think this template should be deleted. Instead, I propose an alternative in [[#Expanding WV:Deletions with Moving to Draft archive]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:22, 9 March 2024 (UTC)

==Expanding WV:Deletions with Moving to Draft archive==
I propose to expand [[WV:Deletion]] with the following:

:&lt;nowiki&gt;&quot;==Moving to Draft archive==&lt;/nowiki&gt;
:&quot;A page that meets the criteria for deletion can be moved to [[Draft:Archive]] instead, unless an overriding rationale for deletion prevails such as the page being offensive, copyright violation, etc. Rationale: The database storage is not saved by deletion and there is generally no harm in being kind to those who hone their writing and wiki editing skills in Wikiversity.&quot;

This matches the practice launched recently by Guy Vandegrift.

This presupposes there will be consensus for this practice.

As a result, we may still use speedy delete and/or rfd process, and handle pages by moving them rather than deleting them. By contrast, this template Draftify introduces a new implied process with unclear rules for what can be moved to Draft space.

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:23, 9 March 2024 (UTC)

:{{Ping|Dan Polansky|Michael Ten}} As I understand it, the plan is to have three options: Delete, Draftspace, or Draft:Archive. If we have three templates, the person who selects the template will have made a ''de facto'' vote on the question.  This could speed up the voting process.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 10:06, 10 March 2024 (UTC)
:: My questions would be these: do you agree with the wording I proposed above? Do you agree that the wording matches the recent practice concerning [[Draft:Archive]] or do you think changes need to be made to the wording I proposed? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:37, 10 March 2024 (UTC)
::Thanks for proposing this, and this seems quite fruitful. I suggested the following amendment or something to this effect (this is just a suggestion and open to constructive changes, unless what you originally proposed is most ideal unmodified). 

:::&quot;==Moving to Draft archive==
:::&quot;A page that meets the criteria for deletion from the main resource namespace that was most likely created in good faith and is not blatant spam should be moved to the &quot;Draft:&quot; namespace instead of being deleted, unless an overriding rationale for deletion prevails such as the page being offensive, copyright violation, spam, and/or so forth. The rational for this is as follows. Not deleting Creating Commons Content created in good faith is fruitful to the Creative Commons as a whole, database storage is not reduced by deleting content, and there is generally no harm moving good faith content to the Draft: namespace. Additionally, whether someone created good faith Creative Commons content (related to teaching, learning, or research) to hone their writing and wiki editing skills, simply to plant the seed of an idea for others to build off of later, or so forth, moving such content to the &quot;Draft:&quot; namespace can give others an opportunity to develop the content later, use the content as food for thought that might spark new, useful, or novel ideas, and/or possibly another creative fruitful intellectual processes related to Creative Commons content creation not described here. 
::Bless up. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 17:32, 10 March 2024 (UTC)
::: I hoped we can agree on something simple, brief and to the point rather than get sidetracked into a discussion about one rationale or the other. And I do not see any substantive defect in my proposal that the above solves. Oh well. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:04, 10 March 2024 (UTC)
::::If I had to choose between the two versions, I would go with the one by [[User:Dan Polansky]]. But I have great respect for the judgement and  vision for Wikiversity expressed by [[User:Michael Ten]].  Also, Dan is fully aware of my personal distaste for the chaotic nature of discussions on all [[w:Wikimedia Foundation|WMF]] wikis.  For that reason, I copied both Dan's and Michael's proposals to '''[[Wikiversity:What-goes-where 2024]]''', and refuse to further discuss it on this insanely long Colloquium page.  Please go to '''[[Wikiversity:What-goes-where 2024]]''' and lets see if we can work this out!--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:01, 11 March 2024 (UTC)
::::: Some edits were made. https://en.wikiversity.org/w/index.php?title=Wikiversity%3ADeletions&amp;diff=2611619&amp;oldid=2605804 [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 03:38, 12 March 2024 (UTC)
::::::I see Jtneill took out some of my sarcasm, which is a good thing to do. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 03:58, 12 March 2024 (UTC)

== How to become paid editor? ==
Many of them edit wikiversity.But is it possible to earn money by editing ? [[User:Musesscab|Musesscab]] ([[User talk:Musesscab|discuss]] • [[Special:Contributions/Musesscab|contribs]]) 14:52, 11 March 2024 (UTC)

:Heads up that I will move this thread once my response is posted. We don't have a local policy page about being a paid editor, but see [[:w:en:Wikipedia:Paid-contribution disclosure]], which references the Wikimedia Foundation's requirements. So that is how to comply with being a paid editor, but how you could ''logistically'' make money editing Wikiversity is an open question. Frankly, I don't think you can. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:24, 11 March 2024 (UTC)
::{{Ping|Musesscab}} I moved this from [[Wikiversity:Requests for Deletion]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:25, 11 March 2024 (UTC)

== Making a Discrete Math course + how can I clean up the math portal? ==

Hi all, I'm considering creating a course on discrete mathematics.  I see in the math portal (https://en.wikiversity.org/wiki/Portal:Mathematics) that there are a number of sub-topics  related to discrete math, but none of them is really a ''course'' on discrete math, as in a unified and sequential presentation like one would get in a university course or textbook on the subject.

I tried digging through the source code to see how things are organized and I got kind of lost in a forest of links and imported content.  I never managed to actually figure out where all the data comes from to make the table of contents.  So I am not clear about how one should go about inserting a new course in the table.  

Moreover, a lot of the stuff that's currently in the table of contents looks kind of disorganized.  I'm not sure if it is intended that single isolated topics would have a whole page dedicated to them -- if not, should they somehow be placed under some kind of larger category?  I'd do it myself, but as noted, I don't see how the organization is built from the source.  

To give a few specific examples: (1) Group theory is under discrete math but that seems like it should get its own high-level category, at least parallel with things like &quot;Geometry&quot;.  (2) Under &quot;Applied mathematics&quot; is &quot;Functional analysis&quot; which seems more than a little wrong (not that Functional Analysis ''can't'' be &quot;applied&quot;, but it is very far from what one would typically call &quot;applied&quot;.).  (3) If anyone wanted to just learn &quot;Discrete mathematics&quot; it's not clear where in that category they should go or start.  Clicking on some things doesn't even really take you to a page but a list of sub-topics, where again one would be naturally confused as to where to start.  

So I don't know if there is a page that I haven't found which would explain this, or if anyone can explain to me how to reorganize the page, or if anyone has other ideas about how to make progress there.  Or, the least effort solution, maybe someone can just make a discrete math course page in whatever way they think is best and then I can fill it in?  

Anyway, I don't want to be a bother but I do look forward to contributing, if anyone would be willing to clear this up!  Thanks.

[[User:Addemf|Addemf]] ([[User talk:Addemf|discuss]] • [[Special:Contributions/Addemf|contribs]]) 16:11, 12 March 2024 (UTC)

:Welcome to Wikiversity! I never dealt with the math portal, but looked at it and made two observations: (1) The portal averages an astonishing 3,000 pageviews per month, and (2) it hasn't been edited since June 2021.  So improving it should be a high priority.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:48, 12 March 2024 (UTC)

: I'd just develop the course in userspace from scratch and then worry about where to put it rather than try to refurbish the existing content like [[Introductory Discrete Mathematics for Computer Science]], which is a bit of a mess and includes dead links. It's somewhat a pet peeve of mine when course material belabors simple points but covers more complicated things only superficially. For instance, why have separate pages for AND, OR, etc, but just a single video (whose link is apparently now dead) about &quot;Proving Programs Correct&quot;? Anyone who's cut out for computer science or math will grasp AND, OR, etc. pretty quickly after glancing at their truth tables and can probably move right along to covering the remainder of propositional logic and working out some exercises. Conversely, most undergrads are probably not going to come away really understanding things like Cantor's diagonal argument or how to apply Floyd–Hoare logic unless significant time is spent on those subjects. A decent discrete math course should spend a good chunk of time on logic (starting with the grammar), sets, proofs, induction, etc and provide a thorough set of exercises. [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 06:01, 28 March 2024 (UTC)
::I tend to agree: Userspace is for individual efforts and draftspace is for collaborative efforts. Perhaps we need to modify {{tl|Welcome}} to emphasize this fact. An exception might be made for individuals working solo on a project, but wish to seek collaborators.  Unfortunately, collaborative efforts are rare-to-nonexistent, unless a classroom instructor is assigning tasks for students to complete on Wikiversity. Seeing non-students to collaborate on a project seems like an unlikely dream.... Incidentally, I cannot ask someone to work in draft-space when we have a regulation stating that draft-space projects get deleted after being dormant for 60 days.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 07:50, 28 March 2024 (UTC)  
::One more thing: {{ping|Addemf}} It is best to get all your work under one top page: In my field, I do everything as a subpage to [[Physics]] (when I first came here it was [[Physics equations]], which I used in the classroom.) That way, when it comes time to move your pages, they all can be moved together. There are so many nut-case pages under [[Physics]] that I created [[Physics/A]], which is an unorthodox solution to the problem. Unfortunately, very few people on Wikiversity are trying to keep their efforts confined in this manner.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 08:14, 28 March 2024 (UTC)
:::&quot;''There are so many nut-case pages under Physics''&quot;. Why not flag them for deletion? [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 09:44, 28 March 2024 (UTC)
::::That is an excellent (though obvious) question! I have two reasons, both involve the same nut-case article: 
{{cot|Collapsed by author Guy vandegrift. The two reasons are bold-faced}}Many years ago, I noticed a page that presented an alternate theory of General Relativity (GR), a theory that uses mathematics so complicated that most people with a Ph.D. in physics never encounter during their education.  Keep in mind that this is not computer science, where old software becomes obsolete: Curvilinear coordinate systems are used to this day, not only in GR, but in plasma physics (where one of the coordinates follows the magnetic field line.) A Wikiversity article on GR attracted my attention because I used my knowledge of curvilinear coordinates to construct a &quot;derivation&quot; of GR that is only a weak field approximation.  I couldn't understand it, but it looked OK (and thankfully wasn't the same as mine.) So the first reason for letting the nut-case pages stay is that '''(1) Wikiversity is not equipped to referee scientific journals.'''  If you look it up, it turns out that the established scientific journals are not very good at refereeing scientific journals. My second reason for leaving the nut-cases alone, is that a Wikipedia editor contacted me about the same article on GR that I looked at. This editor told me that the GR article I looked at but did not understand was impossible.  I was able to google this Wikiversity editor and learned that he had published one or two articles on weak-field GR, so I took their word for it and deleted the article.  This caused a long discussion on [[Wikiversity:Requests for deletions|RFD]] that finally got the page sent to the author's userspace... Then, about the time when Dave was semi-retiring from Wikiversity, I noticed the same page on Wikiversity.  I put it up for RFD (I now realize that I should have unceremoniously deleted it.)  The RFD has recently become so bogged down that only two people responded, namely the nut and I. So, I put it into [[Physics/Essays]] and went on to find easier tasks. So my second reason for not deleting it is '''(2) I gave up on trying to remove low quality pages (there are simply too many of them on WV.)'''   I once read (but no longer believe) that most of the internet is devoted to porn.  But except for the fact that porn might harm children, porn does not prevent the internet from being useful. We have search engines to find the good stuff.  I do delete scientific articles that promote well-known fringe theories because like porn, they are potentially harmful. The bottom line is that I guess Wikiversity should be the place where [https://www.youtube.com/watch?v=tN2KHeEWqZM anything goes]. It's a place where students can write anything that is harmless. &lt;s&gt;Our&lt;/s&gt; ''My'' job is to delete the harmful and separate the harmless from what may or may not (probably the latter) be useful.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:43, 28 March 2024 (UTC)
::::: I'm not sure I understand. If you come across an article that you're sure is bogus you get rid of it (or at least move it to the author's userspace), right? A meaningful contribution to theoretical physics would presumably be a welcome submission in some or other traditional journal, so in any case the stakes seem quite low for theoretical physicists and anyone else doing legitimate work in the hard sciences. [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 15:21, 29 March 2024 (UTC)
::::::{{ping|AP295}} I don't know whether to remove the cot/cob template or not.  But to address (not answer) your question, I was only a participant in introduction of draftspace, and the practice of allowing low quality material to remain in namepage-subspace under titles like [[Student Projects]] and [[Physics/Essays]]. I was focused mostly on using Wikipedia to develop course materials. Now I find myself to be the only active custodian on [[Wikiversity:Requests for Deletion]].  I created [[Wikiversity:WGW2024|What-goes-where 2024]] and am waiting for a consensus to form regarding what the community wishes to do.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 21:42, 29 March 2024 (UTC)
{{cob}}
::I mostly agree, I think there's good stuff in that course that I don't want to disturb, but certainly dead links are always a bad thing.  It's also not ''exactly'' how I'd design it, so I'm making my own.  
::I do think I have different sensibilities about things to focus on.  There are a lot of very fast and technical &quot;reference&quot; books on discrete math and logic, so I don't think anything is really gained by making yet another one.  At least I have nothing to contribute to that.  
::I do think there's value in developing the subject, in some sense, &quot;out of necessity&quot;.  First present a need for something, which a completely uninitiated student can understand the need for.  For example, rather than jump into boolean operators, first present what use they have, and then develop the theory to fit the need.  
::That may mean that I emphasize some things, which a professional will think is trivial. But it is meant as an introduction to the uninitiated and not a reference. [[User:Addemf|Addemf]] ([[User talk:Addemf|discuss]] • [[Special:Contributions/Addemf|contribs]]) 16:10, 28 March 2024 (UTC)
:::{{ping|Addemf}} The worst-case scenario is that you are a student learning to teach mathematics.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:37, 28 March 2024 (UTC)
:::My point is not about emphasis per se, just that one should give each topic the attention and detail it's due. Really just common sense, but more than once I've seen someone make hard work of presenting simple ideas yet omit detail and emphasis where they're needed. Consider also that most undergrads (at least in my limited experience) will quickly forget any non-trivial concept if it's just presented as a one-off. [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 06:57, 29 March 2024 (UTC)

==  Wikimedia Foundation Board of Trustees 2024 Selection ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
: ''[[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Announcement/Selection announcement| You can find this message translated into additional languages on Meta-wiki.]]''
: ''&lt;div class=&quot;plainlinks&quot;&gt;[[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Announcement/Selection announcement|{{int:interlanguage-link-mul}}]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Wikimedia Foundation elections/2024/Announcement/Selection announcement}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]&lt;/div&gt;''
Dear all,

This year, the term of 4 (four) Community- and Affiliate-selected Trustees on the Wikimedia Foundation Board of Trustees will come to an end [1]. The Board invites the whole movement to participate in this year’s selection process and vote to fill those seats.

The [[m:Special:MyLanguage/Wikimedia Foundation elections committee|Elections Committee]] will oversee this process with support from Foundation staff [2]. The Board Governance Committee created a Board Selection Working Group from Trustees who cannot be candidates in the 2024 community- and affiliate-selected trustee selection process composed of Dariusz Jemielniak, Nataliia Tymkiv, Esra'a Al Shafei, Kathy Collins, and Shani Evenstein Sigalov [3]. The group is tasked with providing Board oversight for the 2024 trustee selection process, and for keeping the Board informed. More details on the roles of the Elections Committee, Board, and staff are here [4]. 

Here are the key planned dates:

*  May 2024: Call for candidates and call for questions
*  June 2024: Affiliates vote to shortlist 12 candidates (no shortlisting if 15 or less candidates apply) [5]
*  June-August 2024: Campaign period
*  End of August / beginning of September 2024: Two-week community voting period
*  October–November 2024: Background check of selected candidates
*  Board's Meeting in December 2024: New trustees seated

Learn more about the 2024 selection process - including the detailed timeline, the candidacy process, the campaign rules, and the voter eligibility criteria - on [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|this Meta-wiki page]], and make your plan.

'''Election Volunteers'''

Another way to be involved with the 2024 selection process is to be an Election Volunteer. Election Volunteers are a bridge between the Elections Committee and their respective community. They help ensure their community is represented and mobilize them to vote. Learn more about the program and how to join on this [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Election Volunteers|Meta-wiki page]].

Best regards,

[[m:Special:MyLanguage/User:Pundit|Dariusz Jemielniak]] (Governance Committee Chair, Board Selection Working Group)

[1] https://meta.wikimedia.org/wiki/Special:MyLanguage/Wikimedia_Foundation_elections/2021/Results#Elected

[2] https://foundation.wikimedia.org/wiki/Committee:Elections_Committee_Charter

[3] https://foundation.wikimedia.org/wiki/Minutes:2023-08-15#Governance_Committee

[4] https://meta.wikimedia.org/wiki/Wikimedia_Foundation_elections_committee/Roles

[5] Even though the ideal number is 12 candidates for 4 open seats, the shortlisting process will be triggered if there are more than 15 candidates because the 1-3 candidates that are removed might feel ostracized and it would be a lot of work for affiliates to carry out the shortlisting process to only eliminate 1-3 candidates from the candidate list.&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MPossoupe_(WMF)|MPossoupe_(WMF)]]19:57, 12 March 2024 (UTC)
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== Please vote on whether to allow pages in draftspace to remain indefinitely ==

See '''[[Wikiversity_talk:Drafts#policy_and_page_change_suggestion]]'''. I have been spending a great deal of time attempting to get consensus on deleting articles in mainspace. Your approval of this change in the policy regarding draftspace will make my life much easier. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 05:50, 13 March 2024 (UTC)
*I [https://en.wikiversity.org/w/index.php?title=Wikiversity_talk%3ADrafts&amp;diff=2612888&amp;oldid=2612773 voted] as input for consensus. This seems like it can be a win-win for all interested parties - clean up main namespace, and also preserve good faith content indefinitely in draft namespace until it can be developed enough to be in main-namespace, serves as component of future Creative Commons content, and/or be a catalyst of ideas for future good faith Creative Contributions in Draft namespace and/or main namespace. limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 07:06, 16 March 2024 (UTC)
*I '''support''' allowing pages in draftspace to remain indefinitely.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:00, 16 March 2024 (UTC)

FYI, I created [[Should Wikiversity allow pages in Draft namespace to stay there indefinitely?]], a format that I love. There is no voting there, only arguments/reasoning. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:55, 18 March 2024 (UTC)

:{{ping|Dan Polansky}} I had mixed feelings about Wikidebates, but perhaps that was because most debates don't interest me, either because I don't care or because I already made up my mind.  But on this topic, I care and I am still a bit undecided. Now I see the value of the Wikidebate, and even added an &quot;objection&quot;. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 04:43, 19 March 2024 (UTC)

===More on the draftspace proposal===
[[File:Wikiversity page location flowchart.svg|thumb|Purging draft-archived pages dewikifies incoming links, category statements, and templates. Unpurged versions are viewable via the history.]]
First, a heads-up on two other places this proposal is being discussed.
#[[Wikiversity_talk:Drafts#policy_and_page_change_suggestion]]
#[[Wikiversity_talk:Deletions#Proposed_modifications]]

Also, I started a flowchart to think about, regardless of how this vote goes.  The issue of draftspace is connected to userspace and deletions in general.  The flowchart is one of many ways one might think about this problem: [[User:Mr. Foobar]] writes the page [[Foobar]]. If it is not deleted, it can be moved into either (1) subspace as [[Foobar Plus/Foobar]], (2) userspace as [[User:Mr. Foobar/Foobar]], or (3) draftspace as [[Draft:Foobar]] (suitable if there are multiple authors, such as [[User:Mrs Foobar]] as shown in the flowchart.)  There is a compelling reason for creating a draft-archive space, with pages like [[Draft:Archive/Foobar]].  Unfortunately I forgot to document that reason  and cannot recall it.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:58, 17 March 2024 (UTC)
n 
Status report: Counting the three polling places, I get the impression is that 4 lean towards the proposal to allow draft-space pages to remain indefinitely, while 2 lean against.  I am happy to report that not one of these 6 seem to be stubbornly digging in to defend their positions.  One of the 4 in favor openly admits that they are not qualified to have an opinion. I respect that: ''He who knows not, and knows not that he knows not, is the fool.''--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 05:32, 18 March 2024 (UTC)

I modified the flow chart in two ways: (1) A suggestion to include &quot;soft-deletions&quot; is included, and (2) draft-space items will be purged of incoming links, category statements and template use, but in a way that permits readers to see the unpurged version in the page's history. For more information, visit [[Wikiversity:What-goes-where 2024]].--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 04:12, 21 March 2024 (UTC)
:Thats a nice chart design. What software or tool was used to create the chart? Is the chart still indicating that good faith Creative Commons contributions/creations that are in the draft namespace will remain there indefinitely as long as they relate to learning, teaching, research, and/or education (i.e. within the scope of this wiki)? limitless peace. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 21:07, 26 March 2024 (UTC)
::Two answers: (1) I used the free Inkscape  download, (2) the chart makes no promises; it's a way to look at the problem and contemplate the options we have and what decisions we need to make.  The image also explains why our [[Wikiversity:Requests for Deletion|RFD]] process is dysfunctional: We have too many options.  I am constructing a place to discuss all this.  People who participate in ''[[Wikiversity:What-goes-where 2024]]'' even get a free gift: For some reason pages with the &quot;Wikiversity:&quot; prefix don't offer visual editing.  ''[[Wikiversity:What-goes-where 2024|What-goes-where]]'' is under construction, but if you go there now you can create your own subpage where visual editing is allowed.  I created a &quot;Guest&quot; page where people can try it out before they take the plunge and create that private subpage. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 23:29, 26 March 2024 (UTC)
:::What should we do with this proposal to allow draft-space articles to remain indefinitely?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 20:30, 28 March 2024 (UTC)

== Your wiki will be in read-only soon ==

&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster.

All traffic will switch on '''{{#time:j xg|2024-03-20|en}}'''. The test will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-03-20T14:00|en}} {{#time:H:i e|2024-03-20T14:00}}]'''.

Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-03-20|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. There will be more notifications about this. A banner will be displayed on all wikis 30 minutes before this operation happens. '''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

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== Template:Graph:Chart  not operational - CSV2Chart ==

The template [[w:en:Template:Graph:Chart|Template:Graph:Chart in Wikipedia]] and thus the template  [[Template:Graph:Chart|Template:Graph:Chart in Wikiversity]] is not operational at the moment. So it does not make sense to use [[CSV2Chart]] and [[Template:Graph:Chart|Template:Graph:Chart in Wikiversity]]  in Wikiversity under the current status. SVG files are also editable (e.g. with Open Source tools like [[w:en:Inkscape|Inkscape]]) for multi-language use of diagrams, so that diagrams can be used with SVG format in mathematical learning resources again. Hope that is a feasible workaround for the current status of the [[w:en:Template:Graph:Chart|template Graph:Chart]]. [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 07:15, 19 March 2024 (UTC)

:It is the best thing we have and the best thing we will have for quite a while. See [[phab:T334940]] and [[mw:Extension:Graph/Plans]]. This is a ''multi-year'' issue. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:20, 19 March 2024 (UTC)
:: Shall we keep the syntax of the graphs as they are (maybe wrapped as a comment) in the learning resource and temporarily use the SVG until the diagram problem is solved? Alternative could also be to add a parameter as &quot;svgfallback&quot; to the Graph template and in case of template maintenance or temporary deactivation the SVG fallback image is shown. Not sure if that is a better choice to fix, so that deactivation and activation of a template does not need authoring activities in the learning resources in  Wikiversity and in Wikipedia at all? --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 08:57, 19 March 2024 (UTC)
:::I have no perspective on that; I can just support whatever makes sense to others. I just wanted to give the context that this will not be resolved any time soon. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:39, 19 March 2024 (UTC)
::::Thank you for the information. [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 10:42, 19 March 2024 (UTC)
:::::Added [[w:en:SVG|SVG export]] feature to [[CSV2Chart]] WebApp. [[CSV2Chart]] was initially created to generate Graph:Chart diagrams from CSV data for wikiversity learning resources. [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 17:33, 26 March 2024 (UTC)
: Wikipedia templates using the graph extension show some kind of warning box that the function is disabled. It would be good to have a similar warning box in the English Wikiversity as well. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:11, 19 March 2024 (UTC)
::{{done}} post-importing {{tl|Graphs disabled}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 11:18, 19 March 2024 (UTC)
::: It seems to have no effect on e.g. [[COVID-19/All-cause deaths/Albania]], where I would expect warning boxes to show up in the places where graphs should be. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:31, 19 March 2024 (UTC)
::::Ah, sorry--you are looking for something ''inline'' that displays with the content of the article? There is an equivalent at en.wp f
:::: that as well.r —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:38, 19 March 2024 (UTC)
::::: What I mean is this: in [[W: COVID-19_pandemic_in_Sweden#Statistics]], there is an information box stating &quot;Graphs are unavailable due to technical issues. There is more info on Phabricator and on MediaWiki.org.&quot; And that box seems to be generated by a template that would plot the graph if the extension were functional. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:03, 19 March 2024 (UTC)
::::: I expanded [[:Template:Graph:Chart]] with a notice that graphs are disabled; the notice now appears e.g. in [[COVID-19/All-cause deaths/Albania]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:04, 26 March 2024 (UTC)
: Perhaps something of an aside, I am using Python's matplotlib plotting library instead of the Graph extension, creating SVG files, and it works reasonably well. The data can be embedded in the Python source code. A disadvantage is that one has to upload the SVG file to Commons as a separate editable entity; it is far from as convenient as the Graph extension. On the other hand, the Python code can do additional calculation based on the data and plot the results, e.g. a moving average. A basic use of Python's matplotlib seems to be simple enough for non-programmers; several very simple examples are at [[Wikibooks:Python Programming/matplotlib]] and more are at https://matplotlib.org/stable/tutorials/pyplot.html. Ideally, one places the Python code (with the data embedded) to the Commons-uploaded SVG file so that non-programmers can change the embedded data later without having to significantly touch the non-data part of the code. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:52, 29 March 2024 (UTC)

==Locking Which is the best religion to follow?==
I propose to lock [[Which is the best religion to follow?]] from anonymous IP editing indefinitely, or at least for a year, or at least for a month. The page is likely to attract low-quality edits from anonymous IP editors; and this has already happened. Anyone serious about making good edits can create an account, or can post using IP on the talk page. This measure is specific to this debate (and not all debates); this one is about a highly controversial subject likely to arouse passions and attract various no-so-gifted would-be contributors. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:47, 9 April 2024 (UTC)

:I not only support the proposal, but would probably agree with just about every kind of restriction on a topic as fraught as that.  If it's not intrinsically absurd to try to rigorously analyze a question like that, then at least it requires someone who has demonstrated discipline and credibility.  [[User:Addemf|Addemf]] ([[User talk:Addemf|discuss]] • [[Special:Contributions/Addemf|contribs]]) 21:44, 9 April 2024 (UTC)
:I see some disruptive editing in the history, hence it has been protected short term (already expired) by a colleague. {{Ping|MathXplore}} do you want to weigh in?
:I am reluctant to do indefinite protection, but I am willing to do a longer term, let it expire and then see if we have issues again. If there are unconstructive edits shortly thereafter, lengthen the protection again, rinse and repeat. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:50, 9 April 2024 (UTC)
:: I have no objections to a longer semi-protection. If there are no objections in a week, we should start the new semi-protection term. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:35, 10 April 2024 (UTC)

== Vote now to select members of the first U4C ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024/Announcement – vote opens|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024/Announcement – vote opens}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Dear all,

I am writing to you to let you know the voting period for the Universal Code of Conduct Coordinating Committee (U4C) is open now through May 9, 2024. Read the information on the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024|voting page on Meta-wiki]] to learn more about voting and voter eligibility.

The Universal Code of Conduct Coordinating Committee (U4C) is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community members were invited to submit their applications for the U4C. For more information and the responsibilities of the U4C, please [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Charter|review the U4C Charter]].

Please share this message with members of your community so they can participate as well.

On behalf of the UCoC project team,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 20:20, 25 April 2024 (UTC)
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      <sha1>a9zbgzdy2m6xv34w8gbpom22i4s1m8l</sha1>
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  </page>
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    <title>School talk:Business</title>
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        <id>2888076</id>
      </contributor>
      <minor />
      <comment>Reverted edits by [[Special:Contributions/LETaNKeN|LETaNKeN]] ([[User_talk:LETaNKeN|talk]]) to last version by [[User:Marshallsumter|Marshallsumter]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="9945" xml:space="preserve">{{Schooltalk}}

==Selling==

How is it possible to have a whole section on business and not have a section on the nature of professional selling and what selling is and requires.  The section on marketing doesn't cover it at all and in the real world marketing is not the same as Selling.  Direct sales jobs account for well over 3million jobs in the USA alone.  Selling is the lifeblood of business.  No business will survive without making sales and most businesses fail because they don't make enough sales and run out of cash.

==Economics, Business==
I don't want to offend anyone but people should really understand things before making departments. Yes economics is related to business, but economics is related to almost everything on earth. Economics should remain its own department. In fact, it would make more sense having economics as part of Social Sciences, because it is a science. [[User:Mattrix18|Mattrix18]] 12:39, 3 November 2006 (UTC)
.

==Organization==
I already finalized the 14 departments of the School of Business, so now if anyone wants to add something, it can now be classified in any of the subdepartments. Knowledge Management and Project Management are now in the Management Department, [[User:Humble_Guy| Humble Guy]]

:I expanded some of the departments and edited/merged some others based on the systems set up by recognized universities, namely Harvard, Columbia and University of Florida (where I went to). Please advise if anything need to be changed. [[User:Rayshan|Rayshan]] 18:24, 23 August 2006 (UTC)

===Merger of departments===

I'd like to propose merging Qualitative and Quantitative Methods with either Decision, Risk and Operations and/or Computer and Information Sciences (CIS). I am not aware of a department/school/major at an existing university after that name. Please advise. [[User:Rayshan|Rayshan]] 21:37, 23 August 2006 (UTC)
: Columbia's DRO is one of the more unusual titles for a department that is usually a grab bag and which the major accrediting agency AACSB describes as &quot;Quantitative Methods&quot;.  To be fair, the difference between quant methods and operations management is unclear at many schools -- despite OM payscales being much higher.
: Therefore, I would propose merging the departments into one which captures the main idea and a phrase often in the naming of these departments.  I would call the merged department ''Decision Sciences''.  That fits with what is used at many schools; and, it makes clear what is in (OR/OM, statistics, actuarial science, econometrics, game theory) and what is out (CS, information systems). --[[Special:Contributions/75.34.23.65|75.34.23.65]] 13:06, 29 March 2008 (UTC)

====Entrepreneurship====

Hi. I think there should be a department specializing in entrepreneurship and entrepreneural skills.

[[User:TheBrain|TheBrain]]

:Yes, I merged the Entrepreneurship program with Business School. [[User:Rayshan|Rayshan]] 15:00, 23 August 2006 (UTC)

==Related knowledge==

===MBApedia===

Before learning about this project, I started [http://www.mbapedia.org MBApedia, the free business encyclopedia]. 

The objectives of MBApedia are similar to the Wikiversity, but there's a twist in each of them:
* Academic community: Courses and class notes from business schools.
* Research projects: most of them applied to real business problems.
* Forum: Discussion and analysis of business problems in the media or from contributors.

I think it would be in our best interest to coordinate either joint research projects and courses or even merge.

Anyone interested in the [[Wikiversity:School of Business]] or [http://www.mbapedia.org MBApedia, the free business encyclopedia], please comment on this proposal.

You can also take contents from the MBApedia site, because it is also released under the GNU FDL.

===Wikibooks===

To anyone, pls help write wikibooks so that the school of business can get into gear. [[User:Humble_Guy| Humble Guy]]

==Business Plan==

Hello, I just wanted to prematurely announce that I intend to slowly write a business plan. Please view http://en.wikibooks.org/wiki/Talk:Wikibooks_business_plan_1
I thought this would be a good place to announce this.

- JOsH

:It does not exist yet... [[User:Rayshan|Rayshan]] 21:37, 23 August 2006 (UTC)

==Other comments==

I need help in honing raw business sense and would like to know when this school will be up. Hope it is sooner than later... B

:Um this is an ongoing project, I don't think it's going to be &quot;up&quot; soon, but you can check out the business articles on Wikipedia tho [[User:Rayshan|Rayshan]] 18:24, 23 August 2006 (UTC)

== Question at help desk ==

[[Wikiversity:Help desk#Short term interest rates|Short term interest rates]]&lt;BR&gt;I need a list by country of short-term interest rates. This should be current short-term interet rates or within the past 3 months.&lt;BR&gt;Can you help me get such a list?

Thanks,
Byron Shoulton

[[Business school]]== Wikibooks School of Business ==
I was thinking about adding this section:
&lt;pre&gt;
==Things you can do==
* Over at Wikibooks there is a page [[b:Wikiversity:School_of_Business]]. 
Any useful content from that page needs to be transfered over to Wikiversity 
so the page can be deleted.]] 
&lt;/pre&gt;
However I am not sure about the proper procedure... It seems that the history of the page at Wikibooks should be saved to note past contributors, but also it should not be cluttering up Wikibooks. Is there anyway to transwiki the page history to Wikiversity, but transwiki it in such a way that the the pages can be merge merged, and Wikibooks contributors' contributions are still noted? --[[User:Remi0o|Remi]] 00:18, 22 March 2007 (UTC)

== master - doctoral departments ==

Is the following really necessary? It seems that we could include material at this high of a level at the respective specific departments. i.e. have all levels be contained at Topic:Account... and not have high level content at a separate mba and phd departments... Yes, no? Comment?

===Graduate===
====Masters====
* [[Topic:Master of Accounting|Master of Accounting (M.Acc.)]]
* [[Topic:Master of Business Administration|Master of Business Administration (MBA)]]
* [[Topic:Master of Business Taxation|Master of Business Taxation (MBT)]]
* [[Topic:Master of Financial Technical Analysis|Master of Financial Technical Analysis (MFTA)]]

====Doctoral====
* [[Topic:Doctorate in Business Administration DBA|Doctorate in Business Administration DBA]]
* [[Topic:Accounting Ph.D|Accounting Ph.D]]
* [[Topic:Decision, Risk &amp; Operations Ph.D|Decision, Risk &amp; Operations Ph.D]]
* [[Topic:Finance &amp; Economics Ph.D|Finance &amp; Economics Ph.D]]
* [[Topic:Management Ph.D|Management Ph.D]]
* [[Topic:Marketing Ph.D|Marketing Ph.D]]
* [[Topic:Organizational Behavior Ph.D|Organizational Behavior Ph.D]]

--[[User:Remi|Remi]] 05:57, 18 January 2008 (UTC)

== moving business links to [[Business links]] ==

Perhaps there are enough to warrant this. What do others think? If discussion is absent or does not run counter to this suggestion, I may do so within around five days. [[User:Emesee|Emesee]] 05:07, 9 August 2008 (UTC)

== Submitting Papers (i.e. Original Research) ==

I have a few papers that I have written for my other classes. They aren't anything profound. By Wikipedia standards, they are considered original research and it doesn't belong there. I think these papers would be a quick way to get the information out there. Is there a section to submit papers for peer-view under the school of Business? Is this even worth while? I think it would be to have repository of papers (almost like a journal) where the School of Business users can look at, use, edit, and even reject/accept the paper outright. 

As for these papers, they shouldn't be static nor ever in a &quot;finished&quot; state. I don't think any paper is perfect and can always be improved. At some point the paper may no longer be applicable where new accepted research making it obsolete, and the paper can be retired to an archive. 

This whole thing might be beyond the scope of what wikiversity has intended, but the point in graduate and doctoral studies is to give back the academics.

Any thoughts?   --[[User:Glinos|Glinos]] 12:27, 10 December 2009 (UTC)
:Thanks for the offer and please see [[Wikiversity:Peer review]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])&lt;/small&gt; 20:46, 24 March 2010 (UTC)

== active participants?  ==

Are all of these active participants active? I will paste the list from the non-talk page here. 

&quot;Ray Shan
Prasenjit Paul
Humble Guy
JWSchmidt
Robbi G
Lisa M. Beeson
M. Judd Warren
Missqueenahearts
Perfect Jav
Matt Kinnia&quot;

Are they/you? [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|talk]]) 15:56, 18 December 2012 (UTC)

: I'm none of the above but am active with the following business-related resources
# [[Dominant group/Business]] which is part of my original research into [[Dominant group]],
# [[Dominant group/Economics]], ditto,
# [[:Template:Income|Income]], a template, and research area,
# Monopolistic practices, which is under development, also an original research field,
# [[:Template:Gene project|Gene project]], a template, original research and product development field,
# [[:Template:Repellor vehicle|Repellor vehicle]], ditto, and
# [[Gift economy]], original research resource. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|talk]]) 20:50, 18 December 2012 (UTC)

== A Good Business School ==

A good business school is one that will teach students how to successfully earn money, hopefully in ways that are ethical and beneficial to all humans on Earth. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|talk]]) 16:20, 18 December 2012 (UTC)

: I agree, but it's not easy! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|talk]]) 20:51, 18 December 2012 (UTC)</text>
      <sha1>qlbb77vlsyvrqm9nuissqm0qcirh71k</sha1>
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    <title>Wikiversity:Namespaces</title>
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      <minor />
      <comment>+shortcut</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13262" xml:space="preserve">[[File:Schooldivdeptstructure.png|thumb|right|300px|Figure 1. Hierarchy of Wikiversity schools and topics.&lt;BR&gt;[http://www.archive.org/details/Wikiversity_Reports_WikiProjects Short podcast] about school and topic pages.&lt;BR&gt;See also: [[Wikiversity:Naming conventions]].]]
{{shortcut|WV:NS}}
{{Namespaces}}
{{selfref|For naming conventions for articles and templates see, [[Wikiversity:Naming conventions]].}}
Like all [[wiki]]s using the [[Topic:MediaWiki|MediaWiki]] software, [[Wikiversity]] has built-in [[m:namespace|namespaces]]. The '''main namespace''', where page names have no prefix, is for [[learning resource]]s. There are several additional namespaces where page names all start with a prefix such as &quot;Topic:&quot; or &quot;Wikiversity:&quot;. The other namespaces have special uses that are described in this page. 

The main namespace is for all Wikiversity pages that deal directly with learning (the educational mission of Wikiversity). The main namespace contains pages that are about learning materials (example: [[Hunter-gatherers project]]), learning projects (example: [[Bibliography and Research Methods]]), and research activities (example: [[Bloom clock project]]). 

The other namespaces have special uses that help Wikiversity participants facilitate and organize their collaborative [[Wikiversity:Content development|content development]] efforts.

In addition to the namespaces that are found at [[w:Wikipedia:Namespace|Wikipedia]], Wikiversity  has &quot;school&quot; and &quot;topic&quot; namespaces. Like the &quot;wikiversity&quot; and &quot;portal&quot; namespaces, the &quot;school&quot; and &quot;topic&quot; namespaces are for meta-content; that is, Wikiversity content that is concerned with the management and organization of Wikiversity rather than being directly concerned with learning and the educational activities that take place in the main namespace. Pages in the &quot;school&quot; namespace are concerned with management and organization of large subject areas and large amounts Wikiversity main namespace content (example: [[School:Medicine]]). Wikiversity schools contain and organize groups of more narrowly-focused content development projects in the &quot;Topic:&quot; namespace. The content development projects in the &quot;Topic:&quot; namespace can be referred to as departments, centers, institutes, divisions or whatever is convenient (example: the [[Topic:Accounting|Accounting Department]]). For more information see: [[Wikiversity:Topics]].

'''[[Wikiversity:Content development|Content development projects]]'''. At Wikipedia, editors have found that it is useful to create a special type of meta-page called a [[w:Wikipedia:WikiProject|WikiProject]] page. At Wikipedia such pages all exist with the project namespace and start with the prefix &quot;Wikipedia:WikiProject:&quot; ([[w:Wikipedia:WikiProject Molecular and Cellular Biology|example]]). At Wikiversity, the pages in the &quot;school&quot; and &quot;topic&quot; namespaces can be thought of as special types of &quot;WikiProject&quot; pages that are concerned with creating and organizing learning resources for broad subject areas (&quot;School:&quot; projects) or more narrow academic topics (&quot;Topic:&quot; projects).

The concept of namespaces is similar to [[w:namespace (programming)|programming language namespaces]] and general [[w:namespace|namespace]]s.

==Basic namespaces==
[[Image:Namespacehierarchy.png|thumb|right|400px|'''Figure 2'''. The [[Wikiversity:Namespaces|Wikiversity namespace]] hierarchy. Any department can be 'contained' by several schools. Any learning project can be 'contained' (linked to) by several departments. ]]
The main namespace of Wikiversity is for education-oriented content (see [[Special:Allpages]]). It is the default namespace and page names for Wikiversity pages that are in the main namespace do not use a prefix. Wikiversity allows the creation of subpages for main namespace page, see: [[Wikiversity:Subpages]].&lt;BR&gt;'''101 problem''' - Not all English-speaking countries number courses starting at 101. &quot;[[Wikiversity:Glossary#F|Foo]] 101&quot; means nothing to some people. It is better to say &quot;Introduction to Foo&quot;. 

===Wikiversity:===
All pages in the ''Wikiversity project'' namespace start with the prefix ''Wikiversity:''. The ''Wikiversity namespace'' is a namespace containing pages that provide information about Wikiversity. The page you are now reading is located in the Wikiversity project namespace. An example of another page in the Wikiversity namespace is [[Wikiversity:Main Page]]. Other important pages in the project namespace are the [[Wikiversity:Policies|policy pages]]. '''Note''': the word &quot;project&quot; is used in many ways in the contexts of many different kinds of collaborative wiki projects. This entire website is the &quot;Wikiversity project&quot;. Wikiversity also contains many collaborative &quot;[[Wikiversity:Content development|content development projects]]&quot; in the &quot;School:&quot; and &quot;Topic:&quot; namespaces. In the main namespace, there are many [[Portal:Learning projects|learning projects]] where Wikiversity participants can engage in collaborative learning activities (see Figure 2). See also: [[Portal:Wikiversity]].

===Portal:===
The portal namespace (prefix ''Portal:'') is for reader-oriented directory/guide pages that help readers find Wikiversity [[Learning resource|content]] that is related to a specific subject. It also may contain links to encourage readers to contribute new content to Wikiversity (see, for example, [[Portal:Learning Projects]]. The &quot;Portal:&quot; namespace can be used together with categories. Wikiversity editors can create new categories to unite departments and projects as needed. Each category can have its own portal page in the &quot;Portal:&quot; namespace. Portals are particularly useful for areas of study that do not fit into the more conventional school&gt;division&gt;department hierarchy. Example of a portal: [[Portal:Science]]. See [[Wikiversity:Portal]].

===School:===
{{further|Wikiversity:WikiProject}}
Pages in the '''School: namespace''' can serve as [[Wikiversity:Content development|content development]] projects for broad subject areas in a similar fashion to Wikipedia &quot;[[w:Wikipedia:WikiProject|wikiprojects]]&quot;. The &quot;School&quot; namespace is for organization of the work of Wikiversity participants who are content producers, those who create [[learning resource]]s and participate in [[Wikiversity:Learning projects|learning projects]]. The names of Wikiversity school pages always start with the &quot;School:&quot; prefix. '''Example''': the Wikiversity [[School:Biology|School  of Biology]] organizes many [[Wikiversity:Namespaces#Topic:|topic pages]] such as [[Topic:Cell Biology]]. See: [[Wikiversity:Schools]].

'''School:''' contains community oriented pages for the improvement of resources under a major category.

===Draft:===
{{further|Wikiversity:Drafts}}
The '''Draft:''' namespace is for pages in development that are not yet ready for main space listing. Draft status may be self-selected by users or determined by respected [[Wikiversity:Staff|community members]]. Resources with little content, low quality, or addressing [[Wikipedia:Wikpedia:Fringe theories|fringe theories]] are likely to be placed in the draft namespace until they are more fully developed.

===Topic:===
{{further|Wikiversity:WikiProject|Wikiversity:Topics}}
'''The use of the Topic: namespace was discontinued in 2016, with content moved to either Portal: or main namespace pages.'''

Historically, pages in the [[Wikiversity:Namespaces|Topic: namespace]] served as [[Wikiversity:Content development|content development]] projects for narrow academic topics in a similar fashion to Wikipedia &quot;[[w:Wikipedia:WikiProject|wikiprojects]]&quot;. The &quot;Topic&quot; namespace was for organization of the work of Wikiversity participants who are content producers, those who create [[learning resource]]s and participate in [[Wikiversity:Learning projects|learning projects]]. The names of topic pages always started with the &quot;Topic:&quot; prefix. Example of a topic page: [[Topic:Cell biology]].

'''Topic:''' pages were primarily used as &quot;Departments&quot;, &quot;Centers&quot;, &quot;Divisions&quot; or similar terms desired by those collaborating in that topic area. For example, under &quot;School:Psychology&quot; might have been &quot;Topic:Social psychology&quot; and &quot;Topic:Clinical psychology&quot;. These specific content development pages in the &quot;Topic:&quot; namespace would then be referred to as the Department of Clinical Psychology or the Division of Social Psychology, depending on preference. See: [[Wikiversity:Topics]].

'''Topic:''' may be the equivalent to Wikipedia's taskforces. These are community oriented pages for the improvement of resources.

===User:===
The [[w:User page|user namespace]] (prefix ''User:'') is a namespace with pages for learning about [[Wikiversity:Who are Wikiversity participants?|Wikiversity participants]] and for sharing personal education experiences, for example containing bookmarks to favorite pages. If you see in the [[Special:RecentChanges|RecentChanges]] list that the user named &quot;Jimbo&quot; has updated some pages, this user name is a link not to Jimbo but to [[User:Jimbo]]. (See [[Special:Listusers]]).

===File:===
Pages in the '''File: namespace''' start with the prefix '''File:'''. The &quot;File:&quot; namespace holds image files, sound files, pdf files and other media files), one page for each file, with a link to the image or sound clip itself. See [[Special:ListFiles]]. There are three versions of links to images and sound files:
#&lt;nowiki&gt;[[File:Foobar.jpg]]&lt;/nowiki&gt; will insert the image directly into the page (not for sound files)
#&lt;nowiki&gt;[[Media:Foobar.jpg]]&lt;/nowiki&gt; will make a text link to the image or type of file
#&lt;nowiki&gt;[[:File:Foobar.jpg]]&lt;/nowiki&gt; will make a text link to the file's description page 
'''Media example''': A sound files [[:Image:Troldhaugen.ogg|Example]]. {{audio|Troldhaugen.ogg|Troldhaugen.ogg}}; see [[Wedding-Day at Troldhaugen]] ([[w:Wikipedia:Media help (Ogg)|Help]] with ogg audio play).&lt;BR&gt;'''See''' [[Help:Media]].

===Mediawiki:===
The [[w:MediaWiki namespace|MediaWiki namespace]] (prefix ''MediaWiki:'') is a namespace containing interface texts such as link labels and messages. They are used for adjusting the localisation (i.e. local version) of interface messages without waiting for a new LanguageXx.php file to get installed. Each label and message has a separate page. Pages in this namespace are [[w:this page is protected|protected]] by default. An automatically generated list of all interface messages is at [[Special:Allmessages]].

===Template:===
The [[w:Template namespace|Template namespace]] (formerly part of the MediaWiki namespace) is used to define a standard text which can then be conveniently added within pages, either the text itself at the time of adding, or a reference to the text at the time of viewing the page. The latter way effectively changes all such occurrences of the standard text automatically by just editing the page where the text is defined. These are listed at [[w:Template messages]] and [[w:Navigational templates]]. The majority of this type is not protected.

===Category:===
The [[:Category:Categories|Category]] namespace contains categories of pages, with each displaying a list of pages in that category and optional additional text. [[Wikiversity:Browse]] is a tool to help navigating the Category namespace, which if not well-coordinated leads to problems. But categories are a powerful way to organize effort, unify content, manage projects and maintain Wikiversity. See [[Wikiversity:FAQ/Categorization]] for more information.

===Help:===
Pages in the help namespace provide information about how to edit Wikiversity webpages and participate within the Wikiversity community. See: [[Help:Contents]]. There are also some similar pages in the &quot;Wikiversity:&quot; namespace such as [[Wikiversity:Glossary]] and others in the main namespace such as [[Introduction to Wikiversity]]. Wikiversity also has pages about how to edit other Wikimedia Foundation sister projects, see [[Wikipedia service-learning courses]].

===Collection:===
{{further|Wikiversity:Books|w:en:Wikipedia:Books{{!}}Wikipedia:Books}}
Pages in the collection namespace operate similarly to English Wikipedia's books since both use the [[m:Book tool|Book tool]]. Collections are bundles of resources on a given topic meant to be printed and shared in book form.

==Talk:==
Except for special pages, each namespace has an associated [[w:talk page|talk namespace]]. The talk namespaces are designated by adding ''talk:'' to the normal prefix. For example, the talk namespace associated with the main article namespace has the prefix ''Talk:'', while the talk namespace associated with the user namespace has the prefix ''User talk:''. Most of the pages in the talk namespaces are used to discuss changes to the corresponding page in the associated namespace. Pages in the ''user talk'' namespace are used to leave messages for a particular user. The user talk namespace is special in that, whenever a user's talk page is edited, that user will see a box saying &quot;You have new messages&quot; on every page that they view until they visit their talk page.

==See also==
*[[Help:Namespaces]]
*[[Wikiversity:Namespaces/Proposals for new namespaces|Proposals for new namespaces]]
*[[Wikiversity:Naming conventions]]
*[[Wikiversity:Vision/2009/Namespace reform]]
[[Category:Namespace]]</text>
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      <minor />
      <comment>/* Wikiversity: */ +shortcut</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13284" xml:space="preserve">[[File:Schooldivdeptstructure.png|thumb|right|300px|Figure 1. Hierarchy of Wikiversity schools and topics.&lt;BR&gt;[http://www.archive.org/details/Wikiversity_Reports_WikiProjects Short podcast] about school and topic pages.&lt;BR&gt;See also: [[Wikiversity:Naming conventions]].]]
{{shortcut|WV:NS}}
{{Namespaces}}
{{selfref|For naming conventions for articles and templates see, [[Wikiversity:Naming conventions]].}}
Like all [[wiki]]s using the [[Topic:MediaWiki|MediaWiki]] software, [[Wikiversity]] has built-in [[m:namespace|namespaces]]. The '''main namespace''', where page names have no prefix, is for [[learning resource]]s. There are several additional namespaces where page names all start with a prefix such as &quot;Topic:&quot; or &quot;Wikiversity:&quot;. The other namespaces have special uses that are described in this page. 

The main namespace is for all Wikiversity pages that deal directly with learning (the educational mission of Wikiversity). The main namespace contains pages that are about learning materials (example: [[Hunter-gatherers project]]), learning projects (example: [[Bibliography and Research Methods]]), and research activities (example: [[Bloom clock project]]). 

The other namespaces have special uses that help Wikiversity participants facilitate and organize their collaborative [[Wikiversity:Content development|content development]] efforts.

In addition to the namespaces that are found at [[w:Wikipedia:Namespace|Wikipedia]], Wikiversity  has &quot;school&quot; and &quot;topic&quot; namespaces. Like the &quot;wikiversity&quot; and &quot;portal&quot; namespaces, the &quot;school&quot; and &quot;topic&quot; namespaces are for meta-content; that is, Wikiversity content that is concerned with the management and organization of Wikiversity rather than being directly concerned with learning and the educational activities that take place in the main namespace. Pages in the &quot;school&quot; namespace are concerned with management and organization of large subject areas and large amounts Wikiversity main namespace content (example: [[School:Medicine]]). Wikiversity schools contain and organize groups of more narrowly-focused content development projects in the &quot;Topic:&quot; namespace. The content development projects in the &quot;Topic:&quot; namespace can be referred to as departments, centers, institutes, divisions or whatever is convenient (example: the [[Topic:Accounting|Accounting Department]]). For more information see: [[Wikiversity:Topics]].

'''[[Wikiversity:Content development|Content development projects]]'''. At Wikipedia, editors have found that it is useful to create a special type of meta-page called a [[w:Wikipedia:WikiProject|WikiProject]] page. At Wikipedia such pages all exist with the project namespace and start with the prefix &quot;Wikipedia:WikiProject:&quot; ([[w:Wikipedia:WikiProject Molecular and Cellular Biology|example]]). At Wikiversity, the pages in the &quot;school&quot; and &quot;topic&quot; namespaces can be thought of as special types of &quot;WikiProject&quot; pages that are concerned with creating and organizing learning resources for broad subject areas (&quot;School:&quot; projects) or more narrow academic topics (&quot;Topic:&quot; projects).

The concept of namespaces is similar to [[w:namespace (programming)|programming language namespaces]] and general [[w:namespace|namespace]]s.

==Basic namespaces==
[[Image:Namespacehierarchy.png|thumb|right|400px|'''Figure 2'''. The [[Wikiversity:Namespaces|Wikiversity namespace]] hierarchy. Any department can be 'contained' by several schools. Any learning project can be 'contained' (linked to) by several departments. ]]
The main namespace of Wikiversity is for education-oriented content (see [[Special:Allpages]]). It is the default namespace and page names for Wikiversity pages that are in the main namespace do not use a prefix. Wikiversity allows the creation of subpages for main namespace page, see: [[Wikiversity:Subpages]].&lt;BR&gt;'''101 problem''' - Not all English-speaking countries number courses starting at 101. &quot;[[Wikiversity:Glossary#F|Foo]] 101&quot; means nothing to some people. It is better to say &quot;Introduction to Foo&quot;. 

===Wikiversity:===
{{shortcut|WV:NS/WV}}
All pages in the ''Wikiversity project'' namespace start with the prefix ''Wikiversity:''. The ''Wikiversity namespace'' is a namespace containing pages that provide information about Wikiversity. The page you are now reading is located in the Wikiversity project namespace. An example of another page in the Wikiversity namespace is [[Wikiversity:Main Page]]. Other important pages in the project namespace are the [[Wikiversity:Policies|policy pages]]. '''Note''': the word &quot;project&quot; is used in many ways in the contexts of many different kinds of collaborative wiki projects. This entire website is the &quot;Wikiversity project&quot;. Wikiversity also contains many collaborative &quot;[[Wikiversity:Content development|content development projects]]&quot; in the &quot;School:&quot; and &quot;Topic:&quot; namespaces. In the main namespace, there are many [[Portal:Learning projects|learning projects]] where Wikiversity participants can engage in collaborative learning activities (see Figure 2). See also: [[Portal:Wikiversity]].

===Portal:===
The portal namespace (prefix ''Portal:'') is for reader-oriented directory/guide pages that help readers find Wikiversity [[Learning resource|content]] that is related to a specific subject. It also may contain links to encourage readers to contribute new content to Wikiversity (see, for example, [[Portal:Learning Projects]]. The &quot;Portal:&quot; namespace can be used together with categories. Wikiversity editors can create new categories to unite departments and projects as needed. Each category can have its own portal page in the &quot;Portal:&quot; namespace. Portals are particularly useful for areas of study that do not fit into the more conventional school&gt;division&gt;department hierarchy. Example of a portal: [[Portal:Science]]. See [[Wikiversity:Portal]].

===School:===
{{further|Wikiversity:WikiProject}}
Pages in the '''School: namespace''' can serve as [[Wikiversity:Content development|content development]] projects for broad subject areas in a similar fashion to Wikipedia &quot;[[w:Wikipedia:WikiProject|wikiprojects]]&quot;. The &quot;School&quot; namespace is for organization of the work of Wikiversity participants who are content producers, those who create [[learning resource]]s and participate in [[Wikiversity:Learning projects|learning projects]]. The names of Wikiversity school pages always start with the &quot;School:&quot; prefix. '''Example''': the Wikiversity [[School:Biology|School  of Biology]] organizes many [[Wikiversity:Namespaces#Topic:|topic pages]] such as [[Topic:Cell Biology]]. See: [[Wikiversity:Schools]].

'''School:''' contains community oriented pages for the improvement of resources under a major category.

===Draft:===
{{further|Wikiversity:Drafts}}
The '''Draft:''' namespace is for pages in development that are not yet ready for main space listing. Draft status may be self-selected by users or determined by respected [[Wikiversity:Staff|community members]]. Resources with little content, low quality, or addressing [[Wikipedia:Wikpedia:Fringe theories|fringe theories]] are likely to be placed in the draft namespace until they are more fully developed.

===Topic:===
{{further|Wikiversity:WikiProject|Wikiversity:Topics}}
'''The use of the Topic: namespace was discontinued in 2016, with content moved to either Portal: or main namespace pages.'''

Historically, pages in the [[Wikiversity:Namespaces|Topic: namespace]] served as [[Wikiversity:Content development|content development]] projects for narrow academic topics in a similar fashion to Wikipedia &quot;[[w:Wikipedia:WikiProject|wikiprojects]]&quot;. The &quot;Topic&quot; namespace was for organization of the work of Wikiversity participants who are content producers, those who create [[learning resource]]s and participate in [[Wikiversity:Learning projects|learning projects]]. The names of topic pages always started with the &quot;Topic:&quot; prefix. Example of a topic page: [[Topic:Cell biology]].

'''Topic:''' pages were primarily used as &quot;Departments&quot;, &quot;Centers&quot;, &quot;Divisions&quot; or similar terms desired by those collaborating in that topic area. For example, under &quot;School:Psychology&quot; might have been &quot;Topic:Social psychology&quot; and &quot;Topic:Clinical psychology&quot;. These specific content development pages in the &quot;Topic:&quot; namespace would then be referred to as the Department of Clinical Psychology or the Division of Social Psychology, depending on preference. See: [[Wikiversity:Topics]].

'''Topic:''' may be the equivalent to Wikipedia's taskforces. These are community oriented pages for the improvement of resources.

===User:===
The [[w:User page|user namespace]] (prefix ''User:'') is a namespace with pages for learning about [[Wikiversity:Who are Wikiversity participants?|Wikiversity participants]] and for sharing personal education experiences, for example containing bookmarks to favorite pages. If you see in the [[Special:RecentChanges|RecentChanges]] list that the user named &quot;Jimbo&quot; has updated some pages, this user name is a link not to Jimbo but to [[User:Jimbo]]. (See [[Special:Listusers]]).

===File:===
Pages in the '''File: namespace''' start with the prefix '''File:'''. The &quot;File:&quot; namespace holds image files, sound files, pdf files and other media files), one page for each file, with a link to the image or sound clip itself. See [[Special:ListFiles]]. There are three versions of links to images and sound files:
#&lt;nowiki&gt;[[File:Foobar.jpg]]&lt;/nowiki&gt; will insert the image directly into the page (not for sound files)
#&lt;nowiki&gt;[[Media:Foobar.jpg]]&lt;/nowiki&gt; will make a text link to the image or type of file
#&lt;nowiki&gt;[[:File:Foobar.jpg]]&lt;/nowiki&gt; will make a text link to the file's description page 
'''Media example''': A sound files [[:Image:Troldhaugen.ogg|Example]]. {{audio|Troldhaugen.ogg|Troldhaugen.ogg}}; see [[Wedding-Day at Troldhaugen]] ([[w:Wikipedia:Media help (Ogg)|Help]] with ogg audio play).&lt;BR&gt;'''See''' [[Help:Media]].

===Mediawiki:===
The [[w:MediaWiki namespace|MediaWiki namespace]] (prefix ''MediaWiki:'') is a namespace containing interface texts such as link labels and messages. They are used for adjusting the localisation (i.e. local version) of interface messages without waiting for a new LanguageXx.php file to get installed. Each label and message has a separate page. Pages in this namespace are [[w:this page is protected|protected]] by default. An automatically generated list of all interface messages is at [[Special:Allmessages]].

===Template:===
The [[w:Template namespace|Template namespace]] (formerly part of the MediaWiki namespace) is used to define a standard text which can then be conveniently added within pages, either the text itself at the time of adding, or a reference to the text at the time of viewing the page. The latter way effectively changes all such occurrences of the standard text automatically by just editing the page where the text is defined. These are listed at [[w:Template messages]] and [[w:Navigational templates]]. The majority of this type is not protected.

===Category:===
The [[:Category:Categories|Category]] namespace contains categories of pages, with each displaying a list of pages in that category and optional additional text. [[Wikiversity:Browse]] is a tool to help navigating the Category namespace, which if not well-coordinated leads to problems. But categories are a powerful way to organize effort, unify content, manage projects and maintain Wikiversity. See [[Wikiversity:FAQ/Categorization]] for more information.

===Help:===
Pages in the help namespace provide information about how to edit Wikiversity webpages and participate within the Wikiversity community. See: [[Help:Contents]]. There are also some similar pages in the &quot;Wikiversity:&quot; namespace such as [[Wikiversity:Glossary]] and others in the main namespace such as [[Introduction to Wikiversity]]. Wikiversity also has pages about how to edit other Wikimedia Foundation sister projects, see [[Wikipedia service-learning courses]].

===Collection:===
{{further|Wikiversity:Books|w:en:Wikipedia:Books{{!}}Wikipedia:Books}}
Pages in the collection namespace operate similarly to English Wikipedia's books since both use the [[m:Book tool|Book tool]]. Collections are bundles of resources on a given topic meant to be printed and shared in book form.

==Talk:==
Except for special pages, each namespace has an associated [[w:talk page|talk namespace]]. The talk namespaces are designated by adding ''talk:'' to the normal prefix. For example, the talk namespace associated with the main article namespace has the prefix ''Talk:'', while the talk namespace associated with the user namespace has the prefix ''User talk:''. Most of the pages in the talk namespaces are used to discuss changes to the corresponding page in the associated namespace. Pages in the ''user talk'' namespace are used to leave messages for a particular user. The user talk namespace is special in that, whenever a user's talk page is edited, that user will see a box saying &quot;You have new messages&quot; on every page that they view until they visit their talk page.

==See also==
*[[Help:Namespaces]]
*[[Wikiversity:Namespaces/Proposals for new namespaces|Proposals for new namespaces]]
*[[Wikiversity:Naming conventions]]
*[[Wikiversity:Vision/2009/Namespace reform]]
[[Category:Namespace]]</text>
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      <comment>/* Portal: */ +shortcut</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13310" xml:space="preserve">[[File:Schooldivdeptstructure.png|thumb|right|300px|Figure 1. Hierarchy of Wikiversity schools and topics.&lt;BR&gt;[http://www.archive.org/details/Wikiversity_Reports_WikiProjects Short podcast] about school and topic pages.&lt;BR&gt;See also: [[Wikiversity:Naming conventions]].]]
{{shortcut|WV:NS}}
{{Namespaces}}
{{selfref|For naming conventions for articles and templates see, [[Wikiversity:Naming conventions]].}}
Like all [[wiki]]s using the [[Topic:MediaWiki|MediaWiki]] software, [[Wikiversity]] has built-in [[m:namespace|namespaces]]. The '''main namespace''', where page names have no prefix, is for [[learning resource]]s. There are several additional namespaces where page names all start with a prefix such as &quot;Topic:&quot; or &quot;Wikiversity:&quot;. The other namespaces have special uses that are described in this page. 

The main namespace is for all Wikiversity pages that deal directly with learning (the educational mission of Wikiversity). The main namespace contains pages that are about learning materials (example: [[Hunter-gatherers project]]), learning projects (example: [[Bibliography and Research Methods]]), and research activities (example: [[Bloom clock project]]). 

The other namespaces have special uses that help Wikiversity participants facilitate and organize their collaborative [[Wikiversity:Content development|content development]] efforts.

In addition to the namespaces that are found at [[w:Wikipedia:Namespace|Wikipedia]], Wikiversity  has &quot;school&quot; and &quot;topic&quot; namespaces. Like the &quot;wikiversity&quot; and &quot;portal&quot; namespaces, the &quot;school&quot; and &quot;topic&quot; namespaces are for meta-content; that is, Wikiversity content that is concerned with the management and organization of Wikiversity rather than being directly concerned with learning and the educational activities that take place in the main namespace. Pages in the &quot;school&quot; namespace are concerned with management and organization of large subject areas and large amounts Wikiversity main namespace content (example: [[School:Medicine]]). Wikiversity schools contain and organize groups of more narrowly-focused content development projects in the &quot;Topic:&quot; namespace. The content development projects in the &quot;Topic:&quot; namespace can be referred to as departments, centers, institutes, divisions or whatever is convenient (example: the [[Topic:Accounting|Accounting Department]]). For more information see: [[Wikiversity:Topics]].

'''[[Wikiversity:Content development|Content development projects]]'''. At Wikipedia, editors have found that it is useful to create a special type of meta-page called a [[w:Wikipedia:WikiProject|WikiProject]] page. At Wikipedia such pages all exist with the project namespace and start with the prefix &quot;Wikipedia:WikiProject:&quot; ([[w:Wikipedia:WikiProject Molecular and Cellular Biology|example]]). At Wikiversity, the pages in the &quot;school&quot; and &quot;topic&quot; namespaces can be thought of as special types of &quot;WikiProject&quot; pages that are concerned with creating and organizing learning resources for broad subject areas (&quot;School:&quot; projects) or more narrow academic topics (&quot;Topic:&quot; projects).

The concept of namespaces is similar to [[w:namespace (programming)|programming language namespaces]] and general [[w:namespace|namespace]]s.

==Basic namespaces==
[[Image:Namespacehierarchy.png|thumb|right|400px|'''Figure 2'''. The [[Wikiversity:Namespaces|Wikiversity namespace]] hierarchy. Any department can be 'contained' by several schools. Any learning project can be 'contained' (linked to) by several departments. ]]
The main namespace of Wikiversity is for education-oriented content (see [[Special:Allpages]]). It is the default namespace and page names for Wikiversity pages that are in the main namespace do not use a prefix. Wikiversity allows the creation of subpages for main namespace page, see: [[Wikiversity:Subpages]].&lt;BR&gt;'''101 problem''' - Not all English-speaking countries number courses starting at 101. &quot;[[Wikiversity:Glossary#F|Foo]] 101&quot; means nothing to some people. It is better to say &quot;Introduction to Foo&quot;. 

===Wikiversity:===
{{shortcut|WV:NS/WV}}
All pages in the ''Wikiversity project'' namespace start with the prefix ''Wikiversity:''. The ''Wikiversity namespace'' is a namespace containing pages that provide information about Wikiversity. The page you are now reading is located in the Wikiversity project namespace. An example of another page in the Wikiversity namespace is [[Wikiversity:Main Page]]. Other important pages in the project namespace are the [[Wikiversity:Policies|policy pages]]. '''Note''': the word &quot;project&quot; is used in many ways in the contexts of many different kinds of collaborative wiki projects. This entire website is the &quot;Wikiversity project&quot;. Wikiversity also contains many collaborative &quot;[[Wikiversity:Content development|content development projects]]&quot; in the &quot;School:&quot; and &quot;Topic:&quot; namespaces. In the main namespace, there are many [[Portal:Learning projects|learning projects]] where Wikiversity participants can engage in collaborative learning activities (see Figure 2). See also: [[Portal:Wikiversity]].

===Portal:===
{{shortcut|WV:NS/PORTAL}}
The portal namespace (prefix ''Portal:'') is for reader-oriented directory/guide pages that help readers find Wikiversity [[Learning resource|content]] that is related to a specific subject. It also may contain links to encourage readers to contribute new content to Wikiversity (see, for example, [[Portal:Learning Projects]]. The &quot;Portal:&quot; namespace can be used together with categories. Wikiversity editors can create new categories to unite departments and projects as needed. Each category can have its own portal page in the &quot;Portal:&quot; namespace. Portals are particularly useful for areas of study that do not fit into the more conventional school&gt;division&gt;department hierarchy. Example of a portal: [[Portal:Science]]. See [[Wikiversity:Portal]].

===School:===
{{further|Wikiversity:WikiProject}}
Pages in the '''School: namespace''' can serve as [[Wikiversity:Content development|content development]] projects for broad subject areas in a similar fashion to Wikipedia &quot;[[w:Wikipedia:WikiProject|wikiprojects]]&quot;. The &quot;School&quot; namespace is for organization of the work of Wikiversity participants who are content producers, those who create [[learning resource]]s and participate in [[Wikiversity:Learning projects|learning projects]]. The names of Wikiversity school pages always start with the &quot;School:&quot; prefix. '''Example''': the Wikiversity [[School:Biology|School  of Biology]] organizes many [[Wikiversity:Namespaces#Topic:|topic pages]] such as [[Topic:Cell Biology]]. See: [[Wikiversity:Schools]].

'''School:''' contains community oriented pages for the improvement of resources under a major category.

===Draft:===
{{further|Wikiversity:Drafts}}
The '''Draft:''' namespace is for pages in development that are not yet ready for main space listing. Draft status may be self-selected by users or determined by respected [[Wikiversity:Staff|community members]]. Resources with little content, low quality, or addressing [[Wikipedia:Wikpedia:Fringe theories|fringe theories]] are likely to be placed in the draft namespace until they are more fully developed.

===Topic:===
{{further|Wikiversity:WikiProject|Wikiversity:Topics}}
'''The use of the Topic: namespace was discontinued in 2016, with content moved to either Portal: or main namespace pages.'''

Historically, pages in the [[Wikiversity:Namespaces|Topic: namespace]] served as [[Wikiversity:Content development|content development]] projects for narrow academic topics in a similar fashion to Wikipedia &quot;[[w:Wikipedia:WikiProject|wikiprojects]]&quot;. The &quot;Topic&quot; namespace was for organization of the work of Wikiversity participants who are content producers, those who create [[learning resource]]s and participate in [[Wikiversity:Learning projects|learning projects]]. The names of topic pages always started with the &quot;Topic:&quot; prefix. Example of a topic page: [[Topic:Cell biology]].

'''Topic:''' pages were primarily used as &quot;Departments&quot;, &quot;Centers&quot;, &quot;Divisions&quot; or similar terms desired by those collaborating in that topic area. For example, under &quot;School:Psychology&quot; might have been &quot;Topic:Social psychology&quot; and &quot;Topic:Clinical psychology&quot;. These specific content development pages in the &quot;Topic:&quot; namespace would then be referred to as the Department of Clinical Psychology or the Division of Social Psychology, depending on preference. See: [[Wikiversity:Topics]].

'''Topic:''' may be the equivalent to Wikipedia's taskforces. These are community oriented pages for the improvement of resources.

===User:===
The [[w:User page|user namespace]] (prefix ''User:'') is a namespace with pages for learning about [[Wikiversity:Who are Wikiversity participants?|Wikiversity participants]] and for sharing personal education experiences, for example containing bookmarks to favorite pages. If you see in the [[Special:RecentChanges|RecentChanges]] list that the user named &quot;Jimbo&quot; has updated some pages, this user name is a link not to Jimbo but to [[User:Jimbo]]. (See [[Special:Listusers]]).

===File:===
Pages in the '''File: namespace''' start with the prefix '''File:'''. The &quot;File:&quot; namespace holds image files, sound files, pdf files and other media files), one page for each file, with a link to the image or sound clip itself. See [[Special:ListFiles]]. There are three versions of links to images and sound files:
#&lt;nowiki&gt;[[File:Foobar.jpg]]&lt;/nowiki&gt; will insert the image directly into the page (not for sound files)
#&lt;nowiki&gt;[[Media:Foobar.jpg]]&lt;/nowiki&gt; will make a text link to the image or type of file
#&lt;nowiki&gt;[[:File:Foobar.jpg]]&lt;/nowiki&gt; will make a text link to the file's description page 
'''Media example''': A sound files [[:Image:Troldhaugen.ogg|Example]]. {{audio|Troldhaugen.ogg|Troldhaugen.ogg}}; see [[Wedding-Day at Troldhaugen]] ([[w:Wikipedia:Media help (Ogg)|Help]] with ogg audio play).&lt;BR&gt;'''See''' [[Help:Media]].

===Mediawiki:===
The [[w:MediaWiki namespace|MediaWiki namespace]] (prefix ''MediaWiki:'') is a namespace containing interface texts such as link labels and messages. They are used for adjusting the localisation (i.e. local version) of interface messages without waiting for a new LanguageXx.php file to get installed. Each label and message has a separate page. Pages in this namespace are [[w:this page is protected|protected]] by default. An automatically generated list of all interface messages is at [[Special:Allmessages]].

===Template:===
The [[w:Template namespace|Template namespace]] (formerly part of the MediaWiki namespace) is used to define a standard text which can then be conveniently added within pages, either the text itself at the time of adding, or a reference to the text at the time of viewing the page. The latter way effectively changes all such occurrences of the standard text automatically by just editing the page where the text is defined. These are listed at [[w:Template messages]] and [[w:Navigational templates]]. The majority of this type is not protected.

===Category:===
The [[:Category:Categories|Category]] namespace contains categories of pages, with each displaying a list of pages in that category and optional additional text. [[Wikiversity:Browse]] is a tool to help navigating the Category namespace, which if not well-coordinated leads to problems. But categories are a powerful way to organize effort, unify content, manage projects and maintain Wikiversity. See [[Wikiversity:FAQ/Categorization]] for more information.

===Help:===
Pages in the help namespace provide information about how to edit Wikiversity webpages and participate within the Wikiversity community. See: [[Help:Contents]]. There are also some similar pages in the &quot;Wikiversity:&quot; namespace such as [[Wikiversity:Glossary]] and others in the main namespace such as [[Introduction to Wikiversity]]. Wikiversity also has pages about how to edit other Wikimedia Foundation sister projects, see [[Wikipedia service-learning courses]].

===Collection:===
{{further|Wikiversity:Books|w:en:Wikipedia:Books{{!}}Wikipedia:Books}}
Pages in the collection namespace operate similarly to English Wikipedia's books since both use the [[m:Book tool|Book tool]]. Collections are bundles of resources on a given topic meant to be printed and shared in book form.

==Talk:==
Except for special pages, each namespace has an associated [[w:talk page|talk namespace]]. The talk namespaces are designated by adding ''talk:'' to the normal prefix. For example, the talk namespace associated with the main article namespace has the prefix ''Talk:'', while the talk namespace associated with the user namespace has the prefix ''User talk:''. Most of the pages in the talk namespaces are used to discuss changes to the corresponding page in the associated namespace. Pages in the ''user talk'' namespace are used to leave messages for a particular user. The user talk namespace is special in that, whenever a user's talk page is edited, that user will see a box saying &quot;You have new messages&quot; on every page that they view until they visit their talk page.

==See also==
*[[Help:Namespaces]]
*[[Wikiversity:Namespaces/Proposals for new namespaces|Proposals for new namespaces]]
*[[Wikiversity:Naming conventions]]
*[[Wikiversity:Vision/2009/Namespace reform]]
[[Category:Namespace]]</text>
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      <id>2622962</id>
      <parentid>2622960</parentid>
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      <text bytes="13331" xml:space="preserve">[[File:Schooldivdeptstructure.png|thumb|right|300px|Figure 1. Hierarchy of Wikiversity schools and topics.&lt;BR&gt;[http://www.archive.org/details/Wikiversity_Reports_WikiProjects Short podcast] about school and topic pages.&lt;BR&gt;See also: [[Wikiversity:Naming conventions]].]]
{{shortcut|WV:NS}}
{{Namespaces}}
{{selfref|For naming conventions for articles and templates see, [[Wikiversity:Naming conventions]].}}
Like all [[wiki]]s using the [[Topic:MediaWiki|MediaWiki]] software, [[Wikiversity]] has built-in [[m:namespace|namespaces]]. The '''main namespace''', where page names have no prefix, is for [[learning resource]]s. There are several additional namespaces where page names all start with a prefix such as &quot;Topic:&quot; or &quot;Wikiversity:&quot;. The other namespaces have special uses that are described in this page. 

The main namespace is for all Wikiversity pages that deal directly with learning (the educational mission of Wikiversity). The main namespace contains pages that are about learning materials (example: [[Hunter-gatherers project]]), learning projects (example: [[Bibliography and Research Methods]]), and research activities (example: [[Bloom clock project]]). 

The other namespaces have special uses that help Wikiversity participants facilitate and organize their collaborative [[Wikiversity:Content development|content development]] efforts.

In addition to the namespaces that are found at [[w:Wikipedia:Namespace|Wikipedia]], Wikiversity  has &quot;school&quot; and &quot;topic&quot; namespaces. Like the &quot;wikiversity&quot; and &quot;portal&quot; namespaces, the &quot;school&quot; and &quot;topic&quot; namespaces are for meta-content; that is, Wikiversity content that is concerned with the management and organization of Wikiversity rather than being directly concerned with learning and the educational activities that take place in the main namespace. Pages in the &quot;school&quot; namespace are concerned with management and organization of large subject areas and large amounts Wikiversity main namespace content (example: [[School:Medicine]]). Wikiversity schools contain and organize groups of more narrowly-focused content development projects in the &quot;Topic:&quot; namespace. The content development projects in the &quot;Topic:&quot; namespace can be referred to as departments, centers, institutes, divisions or whatever is convenient (example: the [[Topic:Accounting|Accounting Department]]). For more information see: [[Wikiversity:Topics]].

'''[[Wikiversity:Content development|Content development projects]]'''. At Wikipedia, editors have found that it is useful to create a special type of meta-page called a [[w:Wikipedia:WikiProject|WikiProject]] page. At Wikipedia such pages all exist with the project namespace and start with the prefix &quot;Wikipedia:WikiProject:&quot; ([[w:Wikipedia:WikiProject Molecular and Cellular Biology|example]]). At Wikiversity, the pages in the &quot;school&quot; and &quot;topic&quot; namespaces can be thought of as special types of &quot;WikiProject&quot; pages that are concerned with creating and organizing learning resources for broad subject areas (&quot;School:&quot; projects) or more narrow academic topics (&quot;Topic:&quot; projects).

The concept of namespaces is similar to [[w:namespace (programming)|programming language namespaces]] and general [[w:namespace|namespace]]s.

==Basic namespaces==
[[Image:Namespacehierarchy.png|thumb|right|400px|'''Figure 2'''. The [[Wikiversity:Namespaces|Wikiversity namespace]] hierarchy. Any department can be 'contained' by several schools. Any learning project can be 'contained' (linked to) by several departments. ]]
The main namespace of Wikiversity is for education-oriented content (see [[Special:Allpages]]). It is the default namespace and page names for Wikiversity pages that are in the main namespace do not use a prefix. Wikiversity allows the creation of subpages for main namespace page, see: [[Wikiversity:Subpages]].&lt;BR&gt;'''101 problem''' - Not all English-speaking countries number courses starting at 101. &quot;[[Wikiversity:Glossary#F|Foo]] 101&quot; means nothing to some people. It is better to say &quot;Introduction to Foo&quot;. 

===Wikiversity:===
{{shortcut|WV:NS/WV}}
All pages in the ''Wikiversity project'' namespace start with the prefix ''Wikiversity:''. The ''Wikiversity namespace'' is a namespace containing pages that provide information about Wikiversity. The page you are now reading is located in the Wikiversity project namespace. An example of another page in the Wikiversity namespace is [[Wikiversity:Main Page]]. Other important pages in the project namespace are the [[Wikiversity:Policies|policy pages]]. '''Note''': the word &quot;project&quot; is used in many ways in the contexts of many different kinds of collaborative wiki projects. This entire website is the &quot;Wikiversity project&quot;. Wikiversity also contains many collaborative &quot;[[Wikiversity:Content development|content development projects]]&quot; in the &quot;School:&quot; and &quot;Topic:&quot; namespaces. In the main namespace, there are many [[Portal:Learning projects|learning projects]] where Wikiversity participants can engage in collaborative learning activities (see Figure 2). See also: [[Portal:Wikiversity]].

===Portal:===
{{shortcut|WV:NS/PORTAL}}
The portal namespace (prefix ''Portal:'') is for reader-oriented directory/guide pages that help readers find Wikiversity [[Learning resource|content]] that is related to a specific subject. It also may contain links to encourage readers to contribute new content to Wikiversity (see, for example, [[Portal:Learning Projects]]. The &quot;Portal:&quot; namespace can be used together with categories. Wikiversity editors can create new categories to unite departments and projects as needed. Each category can have its own portal page in the &quot;Portal:&quot; namespace. Portals are particularly useful for areas of study that do not fit into the more conventional school&gt;division&gt;department hierarchy. Example of a portal: [[Portal:Science]]. See [[Wikiversity:Portal]].

===School:===
{{further|Wikiversity:WikiProject}}
Pages in the '''School: namespace''' can serve as [[Wikiversity:Content development|content development]] projects for broad subject areas in a similar fashion to Wikipedia &quot;[[w:Wikipedia:WikiProject|wikiprojects]]&quot;. The &quot;School&quot; namespace is for organization of the work of Wikiversity participants who are content producers, those who create [[learning resource]]s and participate in [[Wikiversity:Learning projects|learning projects]]. The names of Wikiversity school pages always start with the &quot;School:&quot; prefix. '''Example''': the Wikiversity [[School:Biology|School  of Biology]] organizes many [[Wikiversity:Namespaces#Topic:|topic pages]] such as [[Topic:Cell Biology]]. See: [[Wikiversity:Schools]].

'''School:''' contains community oriented pages for the improvement of resources under a major category.

===Draft:===
{{shortcut|WV:NS/D}}
{{further|Wikiversity:Drafts}}
The '''Draft:''' namespace is for pages in development that are not yet ready for main space listing. Draft status may be self-selected by users or determined by respected [[Wikiversity:Staff|community members]]. Resources with little content, low quality, or addressing [[Wikipedia:Wikpedia:Fringe theories|fringe theories]] are likely to be placed in the draft namespace until they are more fully developed.

===Topic:===
{{further|Wikiversity:WikiProject|Wikiversity:Topics}}
'''The use of the Topic: namespace was discontinued in 2016, with content moved to either Portal: or main namespace pages.'''

Historically, pages in the [[Wikiversity:Namespaces|Topic: namespace]] served as [[Wikiversity:Content development|content development]] projects for narrow academic topics in a similar fashion to Wikipedia &quot;[[w:Wikipedia:WikiProject|wikiprojects]]&quot;. The &quot;Topic&quot; namespace was for organization of the work of Wikiversity participants who are content producers, those who create [[learning resource]]s and participate in [[Wikiversity:Learning projects|learning projects]]. The names of topic pages always started with the &quot;Topic:&quot; prefix. Example of a topic page: [[Topic:Cell biology]].

'''Topic:''' pages were primarily used as &quot;Departments&quot;, &quot;Centers&quot;, &quot;Divisions&quot; or similar terms desired by those collaborating in that topic area. For example, under &quot;School:Psychology&quot; might have been &quot;Topic:Social psychology&quot; and &quot;Topic:Clinical psychology&quot;. These specific content development pages in the &quot;Topic:&quot; namespace would then be referred to as the Department of Clinical Psychology or the Division of Social Psychology, depending on preference. See: [[Wikiversity:Topics]].

'''Topic:''' may be the equivalent to Wikipedia's taskforces. These are community oriented pages for the improvement of resources.

===User:===
The [[w:User page|user namespace]] (prefix ''User:'') is a namespace with pages for learning about [[Wikiversity:Who are Wikiversity participants?|Wikiversity participants]] and for sharing personal education experiences, for example containing bookmarks to favorite pages. If you see in the [[Special:RecentChanges|RecentChanges]] list that the user named &quot;Jimbo&quot; has updated some pages, this user name is a link not to Jimbo but to [[User:Jimbo]]. (See [[Special:Listusers]]).

===File:===
Pages in the '''File: namespace''' start with the prefix '''File:'''. The &quot;File:&quot; namespace holds image files, sound files, pdf files and other media files), one page for each file, with a link to the image or sound clip itself. See [[Special:ListFiles]]. There are three versions of links to images and sound files:
#&lt;nowiki&gt;[[File:Foobar.jpg]]&lt;/nowiki&gt; will insert the image directly into the page (not for sound files)
#&lt;nowiki&gt;[[Media:Foobar.jpg]]&lt;/nowiki&gt; will make a text link to the image or type of file
#&lt;nowiki&gt;[[:File:Foobar.jpg]]&lt;/nowiki&gt; will make a text link to the file's description page 
'''Media example''': A sound files [[:Image:Troldhaugen.ogg|Example]]. {{audio|Troldhaugen.ogg|Troldhaugen.ogg}}; see [[Wedding-Day at Troldhaugen]] ([[w:Wikipedia:Media help (Ogg)|Help]] with ogg audio play).&lt;BR&gt;'''See''' [[Help:Media]].

===Mediawiki:===
The [[w:MediaWiki namespace|MediaWiki namespace]] (prefix ''MediaWiki:'') is a namespace containing interface texts such as link labels and messages. They are used for adjusting the localisation (i.e. local version) of interface messages without waiting for a new LanguageXx.php file to get installed. Each label and message has a separate page. Pages in this namespace are [[w:this page is protected|protected]] by default. An automatically generated list of all interface messages is at [[Special:Allmessages]].

===Template:===
The [[w:Template namespace|Template namespace]] (formerly part of the MediaWiki namespace) is used to define a standard text which can then be conveniently added within pages, either the text itself at the time of adding, or a reference to the text at the time of viewing the page. The latter way effectively changes all such occurrences of the standard text automatically by just editing the page where the text is defined. These are listed at [[w:Template messages]] and [[w:Navigational templates]]. The majority of this type is not protected.

===Category:===
The [[:Category:Categories|Category]] namespace contains categories of pages, with each displaying a list of pages in that category and optional additional text. [[Wikiversity:Browse]] is a tool to help navigating the Category namespace, which if not well-coordinated leads to problems. But categories are a powerful way to organize effort, unify content, manage projects and maintain Wikiversity. See [[Wikiversity:FAQ/Categorization]] for more information.

===Help:===
Pages in the help namespace provide information about how to edit Wikiversity webpages and participate within the Wikiversity community. See: [[Help:Contents]]. There are also some similar pages in the &quot;Wikiversity:&quot; namespace such as [[Wikiversity:Glossary]] and others in the main namespace such as [[Introduction to Wikiversity]]. Wikiversity also has pages about how to edit other Wikimedia Foundation sister projects, see [[Wikipedia service-learning courses]].

===Collection:===
{{further|Wikiversity:Books|w:en:Wikipedia:Books{{!}}Wikipedia:Books}}
Pages in the collection namespace operate similarly to English Wikipedia's books since both use the [[m:Book tool|Book tool]]. Collections are bundles of resources on a given topic meant to be printed and shared in book form.

==Talk:==
Except for special pages, each namespace has an associated [[w:talk page|talk namespace]]. The talk namespaces are designated by adding ''talk:'' to the normal prefix. For example, the talk namespace associated with the main article namespace has the prefix ''Talk:'', while the talk namespace associated with the user namespace has the prefix ''User talk:''. Most of the pages in the talk namespaces are used to discuss changes to the corresponding page in the associated namespace. Pages in the ''user talk'' namespace are used to leave messages for a particular user. The user talk namespace is special in that, whenever a user's talk page is edited, that user will see a box saying &quot;You have new messages&quot; on every page that they view until they visit their talk page.

==See also==
*[[Help:Namespaces]]
*[[Wikiversity:Namespaces/Proposals for new namespaces|Proposals for new namespaces]]
*[[Wikiversity:Naming conventions]]
*[[Wikiversity:Vision/2009/Namespace reform]]
[[Category:Namespace]]</text>
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  <page>
    <title>Wikiversity:Manual of Style</title>
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      <text bytes="2790" xml:space="preserve">{{Proposed policy|WV:MOS|WV:MoS|WV:Mos}}

The '''Manual of Style''' is a set of guidelines for the formatting of articles on Wikiversity, currently based loosely on [[w:Wikipedia:Manual of Style|Wikipedia's manual of style]].
{{TOCright}}
== Article names ==
{{shortcut|WV:MOS/N}}
* Article names should (generally) be in sentence case (e.g., [[Social psychology]]), although proper names may also be used e.g., [[John Dewey]]. Full capitalisation is appropriate e.g., for acronyms e.g., [[SPSS]].
* Article names should not contain special characters like '''&amp;''', '''|''', '''-''', and '''+''' unless necessary (there are exceptions, such as the C++ programming language, or when using '''/''' to make subpages).
* Spaces should be used appropriately, and will be automatically converted to underscores in URLs. For example, '''Main Page''' and not '''MainPage''' or '''mainPage'''
* Individual [[WV:School|Schools]] may come up with their own guidelines on article names.  
* Where appropriate, the article name may be bolded if it occurs within the first paragraph.
* For more information, see [[Wikiversity:Naming conventions]]

== Headings ==
Liberal use of headings is encouraged. Semicolons can be used but, as they cannot be linked to as sections of an article appropriately, using the equal sign method is preferred. Articles should, where at all possible, be split into subsections of reasonable length, separated by headings surrounded by two equals signs.

=== Syntax ===
Following is an example of the Mediawiki syntax for headings:

 Introductory paragraph.
 
 == Heading ==
 Text.

 == Heading ==
 Text.

Subsections can be created by using three or four equals signs, with four being parsed as a lower heading level than three. For example

 Introductory paragraph.
 
 == Heading ==
 Text.
 
 === Subheading ===
 Text.
 
 === Subheading ===
 Text.
 
 ==== Deeper subheading ====
 Text.
 
 == Heading ==
 Text.

=== Conventions ===
* It is suggested that headings not contain special characters: when a heading is linked to, special characters are converted to a format suitable for URLs, which can confuse visitors.
* Avoid using Heading Level 1, for example &lt;nowiki&gt;= Section name =&lt;/nowiki&gt;, as this causes problems with site navigation and usability. There are very few instances where this is appropriate. If you would like to change the format of the text such as the font size there are other tools to accomplish this.
* Keep headings short: one or two words is often enough.
* Do not repeat titles or significant parts thereof (exceptions include numbering, e.g. appending integers or roman numerals to headings).
* See [[meta:Help:Section]] for more information.

{{Official policies}}
{{Proposed policies}}

[[Category:Style guide]]
[[Category:Wikiversity policy]]</text>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2823" xml:space="preserve">{{Proposed policy|WV:MOS|WV:MoS|WV:Mos}}

The '''Manual of Style''' is a set of guidelines for the formatting of articles on Wikiversity, currently based loosely on [[w:Wikipedia:Manual of Style|Wikipedia's manual of style]].
{{TOCright}}
== Article names ==
{{shortcut|WV:MOS/N}}
* Article names should (generally) be in sentence case (e.g., [[Social psychology]]), although proper names may also be used e.g., [[John Dewey]]. Full capitalisation is appropriate e.g., for acronyms e.g., [[SPSS]].
* Article names should not contain special characters like '''&amp;''', '''|''', '''-''', and '''+''' unless necessary (there are exceptions, such as the C++ programming language, or when using '''/''' to make subpages).
* Spaces should be used appropriately, and will be automatically converted to underscores in URLs. For example, '''Main Page''' and not '''MainPage''' or '''mainPage'''
* Individual [[WV:School|Schools]] may come up with their own guidelines on article names.  
* Where appropriate, the article name may be bolded if it occurs within the first paragraph.
* For more information, see [[Wikiversity:Naming conventions]]

== Headings ==
{{shortcut|WV:MOS/H|WV:HEADING}}
Liberal use of headings is encouraged. Semicolons can be used but, as they cannot be linked to as sections of an article appropriately, using the equal sign method is preferred. Articles should, where at all possible, be split into subsections of reasonable length, separated by headings surrounded by two equals signs.

=== Syntax ===
Following is an example of the Mediawiki syntax for headings:

 Introductory paragraph.
 
 == Heading ==
 Text.

 == Heading ==
 Text.

Subsections can be created by using three or four equals signs, with four being parsed as a lower heading level than three. For example

 Introductory paragraph.
 
 == Heading ==
 Text.
 
 === Subheading ===
 Text.
 
 === Subheading ===
 Text.
 
 ==== Deeper subheading ====
 Text.
 
 == Heading ==
 Text.

=== Conventions ===
* It is suggested that headings not contain special characters: when a heading is linked to, special characters are converted to a format suitable for URLs, which can confuse visitors.
* Avoid using Heading Level 1, for example &lt;nowiki&gt;= Section name =&lt;/nowiki&gt;, as this causes problems with site navigation and usability. There are very few instances where this is appropriate. If you would like to change the format of the text such as the font size there are other tools to accomplish this.
* Keep headings short: one or two words is often enough.
* Do not repeat titles or significant parts thereof (exceptions include numbering, e.g. appending integers or roman numerals to headings).
* See [[meta:Help:Section]] for more information.

{{Official policies}}
{{Proposed policies}}

[[Category:Style guide]]
[[Category:Wikiversity policy]]</text>
      <sha1>2l3wask9ew2oy2c904oyof9bdy7f6ra</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Naming conventions</title>
    <ns>4</ns>
    <id>85035</id>
    <revision>
      <id>2622938</id>
      <parentid>2592676</parentid>
      <timestamp>2024-04-26T02:22:48Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>/* Casing */ +shortcut</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3888" xml:space="preserve">{{proposed|WV:NC}}
{{selfref|For namespace domains see, [[Wikiversity:Namespaces]].}}
These '''naming conventions''' describe useful rules for naming pages. Naming conventions help [[Wikiversity:Who are Wikiversity participants?|Wikiversity participants]] quickly locate and understand the topic of each [[learning resources|learning resource]]. This policy describes Wikiversity's page and page-section naming conventions. For information on how to organize pages see [[Wikiversity:Namespaces]].

== Keep names simple and concise ==
Educators often know more than learners do about a topic. For example, although an educator might know what &quot;''{{w|Vulpes vulpes}}''&quot; means, &quot;red fox&quot; is more likely to be immediately understood and found by the majority of English learners who wish to learn about red foxes. 

By using simple and concise [[Help:Page name|page]] and [[Introduction to Wikiversity#Page section navigation|section]] names that avoid undue expectations, [[Wikiversity:Who are Wikiversity participants?#Learners / Students|learners]] should be able to quickly locate [[learning resources]] and understand what to expect.

It is suggested to title pages with the subject name followed by the descriptor after a comma or between a parenthesis such as:

 Subject, descriptor

''or''

 Subject (descriptor)

Another common temptation is to include course and lesson numbers (&quot;Biology 101&quot;, &quot;Biology/Lesson 2&quot;, etc.) in resource names. However, learners may either have preexisting expectations about the content of numbered resources from their experiences at specific brick and mortar schools, or may be completely unfamiliar with their meaning. For newer projects try to keep chapter or lesson numbers out of names for sub-pages, however, it is acceptable to title the link with a chapter or lesson number.

For older pages it is easier and acceptable to leave them as they are, as trying to fix it might cause broken links.

== Casing ==
{{shortcut|WV:NC/C|WV:CASING}}
You can use title or sentence casing for pages and section names:
* ''Title Case:'' Advanced Multivariable Calculus
* ''Sentence case:'' Advanced multivariable calculus
Please be consistent with all pages that are part of the same course or curriculum. ''Note:'' that Wikipedia uses sentence casing.

== Acronyms and abbreviations ==
Spell out abbreviations and acronyms in page names. Many acronyms have more than one possible meaning and are not universally understood. For example [http://www.acronymfinder.com/USA.html USA] could be &quot;United States of America&quot; or &quot;Union of South Africa&quot;. By avoiding acronyms and abbreviations in page names, learners are more likely to find exactly what they expect.

== English dialects ==
You can use whatever dialect of English you want for page and section names (e.g., [[w:Comparison of American and British English|American English or British English]]), but be consistent. For example, if you use American English spelling for a page name, you should also use American English throughout the page.

==Summary==
# Keep it simple
# Use descriptive titles for the learning resources not lesson codes
# For resources in the main namespace, don't capitalize, so that it can be linked naturally from running texts from other pages

==Namespaces==
{{Further|Wikiversity:Namespaces}}
# '''Wikiversity:''' is the project namespace
# '''Help:''' are for helps with MediaWiki techniques
# '''Portal:''' A portal is a door to Wikiversity; It helps participants explore according to their interests and organize the resources they want. Portals are also equivalent to Wikipedia's Wikiprojects.
# '''School:''' is for cross-department Wikiversity community projects.

== See also ==
* [[Wikiversity:Manual of Style#Article names]]
{{proposed policies}}

[[de:Wikiversity:Namenskonventionen]]
[[fr:Wikiversité:Conventions de nommage]]
[[ja:Wikiversity:項目名の付け方]]</text>
      <sha1>szblp6s309kx0owwknqgl6rtmynftww</sha1>
    </revision>
    <revision>
      <id>2622942</id>
      <parentid>2622938</parentid>
      <timestamp>2024-04-26T02:27:15Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>/* Acronyms and abbreviations */ +shortcut</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3936" xml:space="preserve">{{proposed|WV:NC}}
{{selfref|For namespace domains see, [[Wikiversity:Namespaces]].}}
These '''naming conventions''' describe useful rules for naming pages. Naming conventions help [[Wikiversity:Who are Wikiversity participants?|Wikiversity participants]] quickly locate and understand the topic of each [[learning resources|learning resource]]. This policy describes Wikiversity's page and page-section naming conventions. For information on how to organize pages see [[Wikiversity:Namespaces]].

== Keep names simple and concise ==
Educators often know more than learners do about a topic. For example, although an educator might know what &quot;''{{w|Vulpes vulpes}}''&quot; means, &quot;red fox&quot; is more likely to be immediately understood and found by the majority of English learners who wish to learn about red foxes. 

By using simple and concise [[Help:Page name|page]] and [[Introduction to Wikiversity#Page section navigation|section]] names that avoid undue expectations, [[Wikiversity:Who are Wikiversity participants?#Learners / Students|learners]] should be able to quickly locate [[learning resources]] and understand what to expect.

It is suggested to title pages with the subject name followed by the descriptor after a comma or between a parenthesis such as:

 Subject, descriptor

''or''

 Subject (descriptor)

Another common temptation is to include course and lesson numbers (&quot;Biology 101&quot;, &quot;Biology/Lesson 2&quot;, etc.) in resource names. However, learners may either have preexisting expectations about the content of numbered resources from their experiences at specific brick and mortar schools, or may be completely unfamiliar with their meaning. For newer projects try to keep chapter or lesson numbers out of names for sub-pages, however, it is acceptable to title the link with a chapter or lesson number.

For older pages it is easier and acceptable to leave them as they are, as trying to fix it might cause broken links.

== Casing ==
{{shortcut|WV:NC/C|WV:CASING}}
You can use title or sentence casing for pages and section names:
* ''Title Case:'' Advanced Multivariable Calculus
* ''Sentence case:'' Advanced multivariable calculus
Please be consistent with all pages that are part of the same course or curriculum. ''Note:'' that Wikipedia uses sentence casing.

== Acronyms and abbreviations ==
{{shortcut|WV:NC/A|WV:ACRONYM|WV:ABBREVIATION}}
Spell out abbreviations and acronyms in page names. Many acronyms have more than one possible meaning and are not universally understood. For example [http://www.acronymfinder.com/USA.html USA] could be &quot;United States of America&quot; or &quot;Union of South Africa&quot;. By avoiding acronyms and abbreviations in page names, learners are more likely to find exactly what they expect.

== English dialects ==
You can use whatever dialect of English you want for page and section names (e.g., [[w:Comparison of American and British English|American English or British English]]), but be consistent. For example, if you use American English spelling for a page name, you should also use American English throughout the page.

==Summary==
# Keep it simple
# Use descriptive titles for the learning resources not lesson codes
# For resources in the main namespace, don't capitalize, so that it can be linked naturally from running texts from other pages

==Namespaces==
{{Further|Wikiversity:Namespaces}}
# '''Wikiversity:''' is the project namespace
# '''Help:''' are for helps with MediaWiki techniques
# '''Portal:''' A portal is a door to Wikiversity; It helps participants explore according to their interests and organize the resources they want. Portals are also equivalent to Wikipedia's Wikiprojects.
# '''School:''' is for cross-department Wikiversity community projects.

== See also ==
* [[Wikiversity:Manual of Style#Article names]]
{{proposed policies}}

[[de:Wikiversity:Namenskonventionen]]
[[fr:Wikiversité:Conventions de nommage]]
[[ja:Wikiversity:項目名の付け方]]</text>
      <sha1>mrcxy3699tepd5cv55xiv55n8hsbnc5</sha1>
    </revision>
    <revision>
      <id>2622945</id>
      <parentid>2622942</parentid>
      <timestamp>2024-04-26T02:28:59Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>/* English dialects */ +shortcut</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3968" xml:space="preserve">{{proposed|WV:NC}}
{{selfref|For namespace domains see, [[Wikiversity:Namespaces]].}}
These '''naming conventions''' describe useful rules for naming pages. Naming conventions help [[Wikiversity:Who are Wikiversity participants?|Wikiversity participants]] quickly locate and understand the topic of each [[learning resources|learning resource]]. This policy describes Wikiversity's page and page-section naming conventions. For information on how to organize pages see [[Wikiversity:Namespaces]].

== Keep names simple and concise ==
Educators often know more than learners do about a topic. For example, although an educator might know what &quot;''{{w|Vulpes vulpes}}''&quot; means, &quot;red fox&quot; is more likely to be immediately understood and found by the majority of English learners who wish to learn about red foxes. 

By using simple and concise [[Help:Page name|page]] and [[Introduction to Wikiversity#Page section navigation|section]] names that avoid undue expectations, [[Wikiversity:Who are Wikiversity participants?#Learners / Students|learners]] should be able to quickly locate [[learning resources]] and understand what to expect.

It is suggested to title pages with the subject name followed by the descriptor after a comma or between a parenthesis such as:

 Subject, descriptor

''or''

 Subject (descriptor)

Another common temptation is to include course and lesson numbers (&quot;Biology 101&quot;, &quot;Biology/Lesson 2&quot;, etc.) in resource names. However, learners may either have preexisting expectations about the content of numbered resources from their experiences at specific brick and mortar schools, or may be completely unfamiliar with their meaning. For newer projects try to keep chapter or lesson numbers out of names for sub-pages, however, it is acceptable to title the link with a chapter or lesson number.

For older pages it is easier and acceptable to leave them as they are, as trying to fix it might cause broken links.

== Casing ==
{{shortcut|WV:NC/C|WV:CASING}}
You can use title or sentence casing for pages and section names:
* ''Title Case:'' Advanced Multivariable Calculus
* ''Sentence case:'' Advanced multivariable calculus
Please be consistent with all pages that are part of the same course or curriculum. ''Note:'' that Wikipedia uses sentence casing.

== Acronyms and abbreviations ==
{{shortcut|WV:NC/A|WV:ACRONYM|WV:ABBREVIATION}}
Spell out abbreviations and acronyms in page names. Many acronyms have more than one possible meaning and are not universally understood. For example [http://www.acronymfinder.com/USA.html USA] could be &quot;United States of America&quot; or &quot;Union of South Africa&quot;. By avoiding acronyms and abbreviations in page names, learners are more likely to find exactly what they expect.

== English dialects ==
{{shortcut|WV:NC/D|WV:DIALECT}}
You can use whatever dialect of English you want for page and section names (e.g., [[w:Comparison of American and British English|American English or British English]]), but be consistent. For example, if you use American English spelling for a page name, you should also use American English throughout the page.

==Summary==
# Keep it simple
# Use descriptive titles for the learning resources not lesson codes
# For resources in the main namespace, don't capitalize, so that it can be linked naturally from running texts from other pages

==Namespaces==
{{Further|Wikiversity:Namespaces}}
# '''Wikiversity:''' is the project namespace
# '''Help:''' are for helps with MediaWiki techniques
# '''Portal:''' A portal is a door to Wikiversity; It helps participants explore according to their interests and organize the resources they want. Portals are also equivalent to Wikipedia's Wikiprojects.
# '''School:''' is for cross-department Wikiversity community projects.

== See also ==
* [[Wikiversity:Manual of Style#Article names]]
{{proposed policies}}

[[de:Wikiversity:Namenskonventionen]]
[[fr:Wikiversité:Conventions de nommage]]
[[ja:Wikiversity:項目名の付け方]]</text>
      <sha1>3weku5sdbja9mm1snxcp2i57r1p585j</sha1>
    </revision>
  </page>
  <page>
    <title>Understanding Arithmetic Circuits</title>
    <ns>0</ns>
    <id>139384</id>
    <revision>
      <id>2622877</id>
      <parentid>2622546</parentid>
      <timestamp>2024-04-25T14:04:36Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3018" xml:space="preserve">
== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20211108.pdf|pdf]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20221130.pdf|pdf]]||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|pdf]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|pdf]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20211111.pdf|pdf]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|pdf]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|pdf]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|pdf]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240425.pdf|C]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|pdf]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>cer8ffumjnc20ikyubfif90zqibnxln</sha1>
    </revision>
  </page>
  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
    <id>171005</id>
    <revision>
      <id>2622872</id>
      <parentid>2622544</parentid>
      <timestamp>2024-04-25T14:02:19Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5186" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20240425.pdf|B.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Complex analysis]]</text>
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  <page>
    <title>Haskell programming in plain view</title>
    <ns>0</ns>
    <id>203942</id>
    <revision>
      <id>2622917</id>
      <parentid>2622775</parentid>
      <timestamp>2024-04-25T23:17:57Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6230" xml:space="preserve">
==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

==Applications==
* Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]])
* Bird's Implementation 
:- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]])
:- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]])
:- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]])
:- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]])

&lt;/br&gt;

==Using GHCi==
* Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]])
&lt;/br&gt;

==Using Libraries==
* Library ([[Media:Library.1.A.20170605.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;


==Types==
* Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]])
* TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]])
* Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]])
* Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]])
* Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]])

==Functions==
* Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]])
* Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]])
* Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]])

==Expressions==
* Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]])
* Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]])
* Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20240426.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20240418.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;

==Function Oriented Typeclasses== 
=== Functors ===
* Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]])
* Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]])
* Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]])

=== Applicatives ===
* Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]])
* Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]])
* Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]])
* Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]])

=== Monads I : Background ===
* Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]])
* Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]])
* Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]])
* Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]])
* IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]])
* Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]])

=== Monads II : State Transformer Monads ===
* State Transformer 
: - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]])
: - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]])
: - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]])

* State Monad 
: - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]])
: - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) 
: - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]])

=== Monads III : Mutable State Monads ===
* Mutability Background 
: - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]])
: - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]])
: - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]])
: - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]])
: - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]])

* Mutable Objects
: - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]])
: - Mutable Data Structures  ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]])

* IO Monad 
: - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]])
: - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]])  
: - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]])  

* ST Monad 
: - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]])
: - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]])
: - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]])

=== Monads IV : Reader and Writer Monads ===
* Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]])
* Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]])
* MonadState Class
:: - State &amp; StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]])
* MonadReader Class
:: - Reader &amp; ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]])
* Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]])

=== Monoid ===
* Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]])

=== Arrow ===
* Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]])
&lt;/br&gt;

==Polymorphism==
* Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]])
&lt;/br&gt;

==Concurrent Haskell ==
&lt;/br&gt;


go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://learnyouahaskell.com/introduction Learn you Haskell]
* [http://book.realworldhaskell.org/read/ Real World Haskell]
* [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material]


[[Category:Haskell|programming in plain view]]</text>
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  <page>
    <title>Understanding Fairness</title>
    <ns>0</ns>
    <id>207028</id>
    <revision>
      <id>2622902</id>
      <parentid>2606513</parentid>
      <timestamp>2024-04-25T19:15:45Z</timestamp>
      <contributor>
        <username>Lbeaumont</username>
        <id>278565</id>
      </contributor>
      <comment>/* Further Reading */ Added Bonobo and Originsof Virtue</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21140" xml:space="preserve">==Introduction==
[[File:Wikipedia scale of justice 2.svg|thumb|right|200px|Fairness is Subtle! ]]
We naturally appeal to fairness to avoid or resolve conflict. Unfortunately when conflict emerges it is often difficult for adversaries to agree on what is actually fair. We often hear the complaint &quot;But that's not fair!&quot; Why is this? This course explores various concepts of fairness and helps the student become aware of the several forms that we consider fair.

==Objectives==
The objectives of this course are to:

*Define the concept of fairness,
*Understand our inherent sense of fairness,
*Demonstrate ambiguity inherent in our concepts of fairness,
*Identify various forms of fairness,
*Become aware of our own bias in suggesting what is fair when negotiating or resolving conflict,
*Explore solutions that avoid bias when advocating for fairness.
If you would like to contact the instructor, please [[Special:Emailuser/Lbeaumont | click here to send me an email]] or leave a comment or question on the [[Talk:Understanding_Fairness|discussion page]].
{{TOC right | limit|limit=1}}
{{100%done}} {{Template:non-formal education}} {{By|lbeaumont}}
The course contains many [[w:Hyperlink|hyperlinks]] to further information. Use your judgment and these [[What_Matters/link_following_guidelines|link following guidelines]] to decide when to follow a link, and when to skip over it.
This course is part of the [[Wisdom/Curriculum|Applied Wisdom Curriculum]].

==Origins of Fairness==
Researcher Frans de Waal studies moral behavior in animals. Collaborating with Dr. Sarah Brosnan they conducted a famous experiment that demonstrated capuchin monkeys rejecting unequal pay. In the experiment two capuchins were caged side by side. The first was given the simple task of handing a small rock to the experimenter. For completing the task the first capuchin was rewarded with a slice of cucumber and seemed satisfied. The second capuchin was then given the same task, and rewarded with a grape. Capuchins find grapes much tastier than cucumber slices. When the task was repeated and the first capuchin rewarded with the cucumber slice, the capuchin immediately rejected this unfair reward by throwing it back at the researcher.  You may enjoy seeing video of the experiment in the TED talk “[https://www.ted.com/talks/frans_de_waal_do_animals_have_morals Moral Behavior in Animals]”, Frans de Wall, filmed November 2011.

Apparently fairness demands equal pay for equal work, at least for capuchins, and perhaps even for humans!

==What is Fair?==
Dictionary definitions of “fair” highlight freedom from bias, dishonesty, or injustice.&lt;ref&gt;Dictionary.com entry for “fair”&lt;/ref&gt;  How is this concept applied? Is it sufficient for the rules of the game to be fair, or must the outcome of the process provide each person with their fair share? Is an unfair outcome evidence of unfair rules? 

The 2009 US Supreme Court case [[w:Ricci_v._DeStefano|Ricci v. DeStefano]] provides an opportunity to explore the concept of fairness. In this case eighteen New Haven Connecticut city firefighters, seventeen of whom were white and one of whom was Hispanic, brought suit under Title VII of the Civil Rights Act of 1964 after they had passed the test for promotions to management positions and the city declined to promote them. New Haven officials invalidated the test results because none of the black firefighters scored high enough to be considered for the positions. Because all of the applicants took the same test, the procedure used to identify the best candidates was fair. However, because the outcome of the process resulted in disparate impact to the black firefighters, the distribution of rewards—being chosen for the job—was unfair.  It was argued that the blacks did not get their fair share. The case highlights a natural tension between the [[w:Disparate_treatment|disparate-treatment]] and [[w:Disparate_impact|disparate-impact]] interpretations of the law.

Courts differed in their decisions on this case. The district court ruled in favor of the city, indicating the disparate-impact interpretation prevailed. The second circuit panel upheld that decision on appeal. However, the Supreme Court ruled in favor the defendants, finding there was no disparate-treatment and rejecting the disparate-impact argument in this case. 

Decisions on how to distribute the [[w:September_11th_Victim_Compensation_Fund|September 11th Victim Compensation Fund]] highlight difficulties in understanding the concept of a “fair share”.  Attorney Kenneth Feinburg became responsible for making the decisions on how much each family of a victim would receive from the $7 billion fund created to compensate the families of the victims of the 9/11 terrorist attacks. This highlights the tension between proportional distribution and equal distribution alternatives.  Distributing the same amount to each family would provide an equal distribution of the funds. However, there are good arguments for using a proportional distribution, using various bases for computing the proportions. 

Here are some alternatives:
*larger families get the larger share because there are more people impacted and more people to divide the funds among,
*Younger people get larger shares, because they will be living longer after the tragedy,
*Distribute shares in proportion to the life earnings lost by person who died in the tragedy.

Proportional distribution based on lost earnings was the primary alternative chosen. A stumbling block to settlements was the fact that many of the World Trade Center victims were highly compensated financial professionals. Families of these victims felt the compensation offers were too low, and, had a court considered their case on an individual basis, they would have been awarded much higher amounts. This concern had to be balanced against the time, complications, and risks of pursuing an individual case, and the real possibility that the airlines and their insurers could be bankrupted before being able to pay the claim.

==Three Forms of Fairness==
These cases provide examples of what Johnathan Haidt describes as three forms of fairness.&lt;ref&gt;[http://democracyjournal.org/magazine/28/of-freedom-and-fairness/ Of Freedom and Fairness], Johnathan Haidt&lt;/ref&gt;&lt;ref&gt;[http://righteousmind.com/what-are-the-fairness-buttons/ What are the fairness buttons?], by Johanthan Haidt&lt;/ref&gt;
These are:

'''Procedural Fairness'''—Playing by the same rules—are honest, open and impartial procedures used to decide who got what?

and

'''Distributive Fairness'''—Does everyone get what they deserve?  This has two interpretations:
*'''Equality'''—Equal outcomes, Equality as a result, everyone gets the same reward.
*'''Proportionality'''—Fair share, based on effort expended, impact suffered, or some other criteria. 
The New Haven firefighters were arguing for procedural fairness over distributive fairness. The families of the 9/11 victims were arguing for various forms of distributive fairness, and various interpretations of proportionality.

==Fair Tax Levies==
Politicians, tax payers, free riders, and revenue beneficiaries argue endlessly over the fairness of tax levies. Tax laws have become very complicated, and the need for and use of revenue collected is hotly debated. Here we will limit discussion to [[w:Income_tax_in_the_United_States#Federal_income_tax_rates_for_individuals|federal income tax rates for individuals]] in the United States. Marginal tax rates currently range from 10% for individuals with yearly taxable income below $9,275, to 39% for those with a single taxable income exceeding $415,051. Effective tax rates are typically lower than marginal rates due to various deductions, and range from zero for those with lower incomes to 20.1% for the top 1% of the population. Because the tax rate increases as the taxable amount increases, this is a [[w:Progressive_tax|progressive tax]].

Is this progressive tax levy fair? Since every person is subjected to the same tax laws, on the surface it appears to be ''procedurally fair''. But the tax law is very complicated, it offers many deductions for various reasons and exempts certain types of entities from paying taxes. [[w:Advocacy_group|Advocacy groups]] are able to influence the tax code, by [[w:Lobbying_in_the_United_States|lobbying]] legislators for example, to gain an advantage. Because individuals rarely have the resources required to influence tax laws, the creation of tax laws is not procedurally fair. Uniformly applying laws created under a procedurally unfair process leads to a procedurally unfair result. 

Is the resulting distribution of taxes fair? Because individuals who earn more income pay more taxes, the outcomes (in terms of the total taxes paid) are not equal. Because the tax is progressive, the tax paid is not proportional to the income earned; it is more than would be proportional under a flat tax. What happens if we look at the distribution of money ''retained after paying tax'' rather than the tax levy itself?  Under today’s progressive tax plan the amount retained increases as the amount earned increases. Tax rates would have to become much more steeply progressive to result in equal amounts retained regardless of the amount earned.

Arguing for or against any particular tax plan on the basis of fairness seems unable to provide a clear resolution of the many issues typically raised. It can be coherently argued that the system is or is not procedurally fair. It can be coherently argued that any particular plan results in unfair distributions of the tax burden, or of the retained income. Perhaps another viewpoint can provide better insight.
=== Assignment ===
#Choose some topic related to the [[w:Distribution_of_wealth|distribution of wealth]] to study for this assignment. This might be [[w:Wealth_concentration|wealth concentration]], [[w:Economic_inequality|economic inequality]], [[w:Progressive_tax|progressive tax]], [[w:Wealth_tax|wealth taxes]], inheritance and [[w:Inheritance_tax|estate taxes]], [[w:Old_money|old money]], [[w:Philanthropy|philanthropy]], [[w:Basic_income|universal basic income]], or some other related topic 
#Read the essay [[/Luck, Land, and Legacy/]].
#Propose a policy position related to your chosen distribution of wealth topic that you believe is fair.

==Deep Pragmatism==
Eighteenth century philosopher [[w:Jeremy_Bentham|Jeremy Bentham]] and his successor [[w:John_Stuart_Mill|John Stuart Mill]] introduced the ''utilitarian'' philosophy.  Jeremy Bentham's famous formulation of utilitarianism is known as the “greatest-happiness principle”. It holds that one must always act so as to produce the greatest aggregate happiness among all sentient beings, within reason. In his recently published book ''Moral Tribes: Emotion, Reason, and the Gap Between Us and Them'', Joshua Green revived interest in this archaic philosophy by addressing the many objections that have been raised over the centuries. He suggests using the name “deep pragmatism”, and as he explains, deep pragmatism provides the answers to two essential questions: What really matters? and Who really matters? After an in-depth exploration of these questions, he provides this shorthand answer: “Happiness is what matters, and everyone’s happiness counts the same.”

It is important to recognize that the term “happiness” as it is used here is not a reference to cheap thrills, but is shorthand for a much deeper gratification, perhaps better called [[w:Well-being|well-being]], [[w:Flourishing|flourishing]], or [[w:Eudaimonia|eudaimonia]]. 

The connection between happiness and wealth is complex.
The [[w:Easterlin_paradox|Easterlin Paradox]] holds that, &quot;high incomes do correlate with happiness, but long term, increased income doesn't correlate with increased happiness.&quot;
The economic law of [[w:Marginal_utility#Diminishing_marginal_utility|diminishing marginal utility]] expresses an important element of the relationship. This law recognizes that a few more dollars is more useful to a low-income person than it is to a high-income individual. For example, to a low income person a modest amount of additional money may mean the vital difference between eating a nutritious meal and going hungry, whereas the same amount of additional money may only slightly increase the opulence or convenience available to a high-income person. A consequence of the law of diminishing marginal utility is that income distributions approaching equality theoretically maximize total utility. This is one basis for [[w:Economic_inequality#Social_justice_arguments|social justice arguments]] in favor of sharply progressive tax levies.

Using the principle of deep pragmatism rather than a variety of arguments based on fairness suggests that the most equitable tax would be a sharply progressive tax. Various claims of [[w:Moral_hazard|moral hazard]] and [[w:Incentive|disincentives]] can raise objections to this approach.

== Inequality ==
Appeals to “reduce inequality” and often to “reduce [[w:Economic_inequality|income inequality]]” are often made in pursuit of fairness.&lt;ref&gt;See, for example materials at [http://Inequality.org Inequality.org].&lt;/ref&gt;

In his book ''On Inequality'', philosopher [[w:Harry_Frankfurt|Harry Frankfurt]] notes that income equality can be achieved by arranging that all incomes are ''equally below'' the [[w:Poverty_threshold|poverty line]]. He then states “Economic equality is not, as such, of any particular moral importance; and by the same token, economic inequality is not in itself morally objectionable.”&lt;ref&gt;{{cite book |last=Frankfurt  |first=Harry G.  |date=September 29, 2015 |title=On Inequality |publisher=Princeton University Press |pages=120 |isbn=978-0691167145}} @17 of 103.&lt;/ref&gt; He goes on to say “What is morally important is that each should have enough” and then introduces the “doctrine of sufficiency”&lt;ref&gt;{{cite book |last=Frankfurt  |first=Harry G.  |date=September 29, 2015 |title=On Inequality |publisher=Princeton University Press |pages=120 |isbn=978-0691167145}} @18 of 103.&lt;/ref&gt;  where what is morally important is that everyone has enough money.
== A Fair Price to Pay ==
Luck influences our lives in many ways. We often mistake chance results for outcomes due to skill or incompetence. Rich people may attribute their wealth to hard work, good character, and excellent judgement, while blaming poor people for being lazy and undeserving. The story “[[Understanding_Fairness/fair_enough|fair enough]]” suggests a [[w:Thought_experiment|thought experiment]] than can help us analyze the role luck plays in our lives, and what we might be willing to pay to ensure our ongoing good luck.
=== Assignment ===
#Read the story [[Understanding_Fairness/fair_enough|fair enough]].
#List those factors in your life that resulted from luck. Consider gender, intelligence, talents, race, birth defects, genetic traits, birthplace, parental structure, family structure, inherited wealth, and other factors.
#For each of these factors, decide if you are satisfied with, dissatisfied with, or indifferent to the attributes you were born with.
#What, if anything, would you like to change?
#How much might you be willing to pay to keep your fortunate attributes?
#How much might you regard as fair compensation for continuing to live with your unfortunate characteristics?
#How would a [[w:Basic_income|basic income]] change your life?

==Steps Toward a Fair Resolution==
We can begin to assemble the various arguments explored above into an outline of steps that may result in a fair [[w:Conflict_resolution|resolution]] of a dispute.
#Identify the nature of the [[Transcending Conflict|conflict]]. What are the real interests  of each part in the dispute?&lt;ref&gt;{{cite book |last1=Fisher |first1=Roger |last2=Ury |first2=William L. |date=1981 |title=[[w:Getting_to_Yes|Getting to YES: Negotiating Agreement Without Giving In]] |location=United Kingdom |publisher=Penguin Group |pages=200 |isbn=978-0140157352}}
&lt;/ref&gt; Is this about money, pride, utility, power, prestige, ideology, spite, humiliation, winning, survival, or something else? What is the real [[Problem Finding|problem that needs to be solved]]?&lt;ref&gt;
{{cite book |last1=Gause |first1=Donald C. |last2=Weinberg |first2=Gerald M. |date=March 1, 1990 |title=Are Your Lights On?: How to Figure Out What the Problem Really Is |publisher=Dorset House Publishing Company, Incorporated |pages=176 |isbn=978-0932633163}}&lt;/ref&gt;
#Explore alternatives based on each of the three forms of fairness: procedural, equal distribution, or proportional distribution. Examine reasons for and against each approach. Be explicit about what form of fairness is being analyzed at each stage of the [[Practicing Dialogue|dialogue]]. Recognize that procedural fairness is typically necessary, but not sufficient to ensure distributive fairness. 
#If dialogue carried out in good faith becomes deadlocked because appeals to fairness have become circular, then a new criteria needs to be introduced. If a resolution cannot be arrived at based on appeals to some form of fairness, then use deep pragmatism to resolve the issue. Ask: What really matters? and Who really matters?
#Apply the principle that “Happiness is what matters, and everyone’s happiness counts the same.” Identify each [[w:Stakeholder_analysis|stakeholder]]. Identify what happiness—understood as [[w:Well-being|well-being]] or [[w:Flourishing|flourishing]]—means to each stakeholder in this case. Explore solutions based on this understanding. 
#Test the fairness of various proposals being considered by offering stakeholders the hypothetical opportunity to change places with each of the other stakeholders. Ask each to describe the fairness of the resolution from this new vantage point. Assess the accuracy of these alternative viewpoint descriptions. This is an application of the [[Living the Golden Rule|golden rule]].

==Assignment==
#Choose an issue from the “[[Understanding_Fairness/struggles for fairness|struggles for fairness]]” gallery, or some other fairness issue you would like to study for this assignment.
#Study the issue thoroughly enough to allow you to argue both for and against a variety of resolutions, based on various reasonable assessments of fairness. 
#Follow the “[[Understanding_Fairness#Steps_Toward_a_Fair_Resolution|Steps toward a fair resolution]]” described above to arrive at your proposed resolution.
# If you wish, change places (i.e. assume the role of another stakeholder to the issue) and repeat the previous step to arrive at your proposed resolution from this new point of view. Does the previously proposed solution still seem fair from this new viewpoint?

==Further Reading==
Students interested in learning more about fairness may be interested in the following materials.
*{{Cite book|title=The bonobo and the atheist: in search of humanism among the primates|last=Waal|first=Frans B. M. de|date=2013|publisher=W.W. Norton &amp; Company|isbn=978-0-393-07377-5|location=New York, NY}}
*{{Cite book|title=The origins of virtue: human instincts and the evolution of cooperation|date=1998|publisher=Penguin Books|isbn=978-0-14-026445-6|editor-last=Ridley|editor-first=Matt|series=A Penguin book : science|location=London}}
* {{cite book |last=Greene |first=Joshua |authorlink=w:Joshua_Greene_(psychologist)|date=December 30, 2014 |title=Moral Tribes: Emotion, Reason, and the Gap Between Us and Them  |publisher=Penguin Books |pages=432 |isbn=978-0143126058}} 
* {{cite book |last1=Pickett |first1=Kate |last2=Wilkinson |first2=Richard |date=April 26, 2011 |title=The Spirit Level: Why Greater Equality Makes Societies Stronger  |publisher=Bloomsbury Press |pages=400 |isbn= 978-1608193417}}
* {{cite book |last1=Fisher |first1=Roger |last2=Ury |first2=William L. |date=1981 |title=[[w:Getting_to_Yes|Getting to YES: Negotiating Agreement Without Giving In]]  |url= |location=United Kingdom |publisher=Penguin Group |pages=200 |isbn=978-0140157352}}
* {{cite book |last1=Gause |first1=Donald C. |last2=Weinberg |first2=Gerald M. |date=March 1, 1990 |title=Are Your Lights On?: How to Figure Out What the Problem Really Is |publisher=Dorset House Publishing Company, Incorporated |pages=176 |isbn=978-0932633163}}
*{{cite book |last=Frankfurt  |first=Harry G. |authorlink=w:Harry_Frankfurt |date=September 29, 2015 |title=On Inequality |publisher=Princeton University Press |pages=120 |isbn=978-0691167145}}
* Searching for books on the topic of &quot;Unfair&quot; returns a long list of titles to consider. Also consider searching on &quot;mediation&quot; or &quot;negotiation&quot;.
*(Evaluate the book:  ''The Fairness Instinct: The Robin Hood Mentality and Our Biological Nature'', by L. Sun )
* (Evaluate the book: ''The Moral Underground: How Ordinary Americans Subvert an Unfair Economy'', by Lisa Dodson)

==References==
&lt;references/&gt;
{{Emotional Competency}}
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[[Category:Philosophy]]
[[Category:Life skills]]
[[Category:Peace studies]]
[[Category:Sociology]]
[[Category:Applied Wisdom]]
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[[File:Doré, Gustave - Paradiso Canto 31.jpg|thumb|alt=A man staring into a large swirl of angels and light|God as a point of light in {{W|Paradiso (Dante)|Dante's ''Paradiso''}}, engraving by {{W|Gustave Doré}}.]]

[[File:Does God exist%3F.png|thumb|Visual snapshot of this debate sometime in the past.]]

Questions about the nature of ultimate reality have been asked as long as humans have been conscious. For thousands of years, across thousands of cultures, belief in a supreme being has been more or less common but some have always called into question whether or not God exists or can even be known.

By &quot;God&quot; we mean the metaphysically ultimate being, all-knowing, all-powerful, all-good, timeless, simple and devoid of any anthropomorphic qualities. We do not necessarily mean the Abrahamic God, although these ideas may share some overlap.

So is there a God?

== God exists ==

=== Pro ===

* {{Argument for}} The laws of logic are necessary true propositions. Propositions are real entities, but cannot be physical entities; they are essentially thoughts.&lt;ref&gt;&quot;Propositions are not on paper, in your brain, or anywhere else&quot; [https://www.academia.edu/630293/Propositions_Are_Not_on_Paper_In_Your_Brain_or_Anywhere_Else Direct Paper link]&lt;/ref&gt;{{Clarify}} So the laws of logic are necessarily true thoughts. Since they are true in every possible world, they must exist in every possible world.&lt;ref&gt;&quot;From Necessary Truth to Necessary Existence&quot; [http://www.joshualrasmussen.com/articles/existence.pdf Direct paper link (PDF)]&lt;/ref&gt; But if there are necessarily existent thoughts, there must be a necessarily existent mind; and if there is a necessarily existent mind, there must be a necessarily existent person. A necessarily existent person must be spiritual in nature, because no physical entity exists necessarily. Thus, if there are laws of logic, there must also be a necessarily existent, personal, spiritual being.&lt;ref&gt;{{Cite journal|last=Anderson|first=James N.|last2=Welty|first2=Greg|date=2011|title=The Lord of Noncontradiction: An Argument for God from Logic|url=https://www.semanticscholar.org/paper/The-Lord-of-Noncontradiction%3A-An-Argument-for-God-Anderson-Welty/e0f49c35c70e54174bf201c59f4abfadd223c583?p2df|doi=10.5840/PC201113229}}&lt;/ref&gt;&lt;ref&gt;&quot;A Defense of Theistic Argument from the Law of Non-contradiction&quot;, [https://oaktrust.library.tamu.edu/bitstream/handle/1969.1/157717/NAVARRO-THESIS-2016.pdf?sequence=1&amp;isAllowed=y Direct paper link (PDF)]&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Weaver|first=Christopher G.|title=Why is There Anything?|url=https://www.academia.edu/20392976/Why_is_There_Anything|journal=Two Dozen (or so) Arguments for God: The Plantinga Project}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Rasmussen|first=Joshua|date=2009-01-01|title=From a necessary being to god|url=https://www.academia.edu/630287/From_a_necessary_being_to_god|journal=International journal for philosophy of religion}}&lt;/ref&gt;
** {{Objection}} This is an argument that the mind exists. That is not the question. The ‘spiritual’ is also undefined, and it’s logical connection to a god is not evident
** {{Objection}} This proposition is not correct: &quot;Since they are true in very possible world, they must exist in every possible world&quot;. Why? Because the expression &quot;possible world&quot; has two different meanings. Is an amphibology. In first case, its meaning is: &quot;possible world that we imagine&quot;. In the second case, its meaning is: &quot;possible world that can exist&quot;.
** {{Objection}} Propositions are not thoughts: they are abstract objects existing in the world of abstract objects together with numbers, shapes, mathematical functions, etc.
** {{Objection}} Re: &quot;They are true in every possible world, they [propositions] must exist in every possible world&quot;: Untrue: propositions do not ''exist in'' possible worlds; they exist in the world of abstract objects together with numbers, shapes, mathematical functions, etc. Propositions ''are true in'' possible worlds.
* {{Argument for}} Existence of God has been proven ''a priori'' using higher-order logic and reasonable axioms. Axioms used in research papers to prove are as follows: A1: Self-identity is a positive property, self-difference is not. A2: A property entailed or necessarily entailed by a positive property is positive. A3: The conjunction of any collection of positive properties is positive.&lt;ref&gt;Axioms:[https://arxiv.org/pdf/2001.04701]&lt;br /&gt;A1 Self-identity is a positive property, self-difference is not.&lt;br /&gt;A2 A property entailed or necessarily entailed by a positive property is positive.&lt;br /&gt;A3 The conjunction of any collection of positive properties is positive.&lt;/ref&gt;&lt;ref&gt;&quot;A (Simplified) Supreme Being Necessarily Exists -- Says the Computer!&quot;, [https://arxiv.org/pdf/2001.04701 Direct Paper link (PDF)]&lt;/ref&gt;{{Clarify}}
** {{Objection}} Non-existence of god has been proven a priori using higher-order logic and reasonable axioms.
*** {{Objection}} What axioms and logics are used to prove it? Logics, Arguments, deductions, explanation, and axioms used to prove existence of (simplified) Supreme being are given in paper.&lt;ref&gt;&quot;A (Simplified) Supreme Being Necessarily Exists -- Says the Computer!&quot;, [https://arxiv.org/pdf/2001.04701 Direct Paper link (PDF)]&lt;/ref&gt;
** {{Objection}} Without statement of a specific proof, there is nothing to respond to. A general reader is left with the option of believing that there exists some kind of proof without being able to verify it, or even get a vague idea of what the proof consists in. The above is essentially the &quot;faith&quot; stratagem.
*** {{Objection}} *God* is merely subjective to interpretation. With that logic, you can’t disprove my version of God, so it is correct. 
**** {{Objection}} According to this logic, we should worship the Flying Spaghetti Monster
* {{Argument for}} Various phenomena in the Universe appear to be designed and suggest a designer, God.{{Example}}
** {{Objection}} The characteristics of “appearing to be designed” is vague and subjective. At best, the appearance of design suggests some force or forces of nature that manifest as complexity and pattern. At worst it is an argument based on projected biological traits. In no way does it necessitate a god.
** {{Objection}} Consider Auroras, these phenomena occurs due to scattering of charged particles in atmosphere which exists as of evolutionary phases of Earth. Not designed by anyone. 
*** {{Objection}} Not a good objection. Just because a farmer grows potatoes doesn’t mean that it’s design wasn’t created.
*** {{Objection}} The fact that some phenomena were not designed by God does not imply that no phenomena were designed by God.
**** {{Objection}} You cannot use some things as proof for the existence of God on the basis that they seem designed, while ignoring others which exhibit similar 'designed' characteristics but which are scientifically proven to be natural.
** {{Objection}} Animals and plants do appear to be designed, but Darwinian theory of evolution by natural selection explains how their design-like features originate without divine intervention. Before Darwinian theory, it was difficult to explain the origin of function in living things. One should not conclude from a lack of explanation for a natural phenomenon that no explanation can be found, thunder and disease being examples of phenomena for which explanations were found later.
*** {{Objection}} Darwinian theory by no means proves that life originated without divine intervention. It just shows that the universe behaved someway at a particular point in time. Why it behaves the way it behaves may have been predetermined by divine intervention.
*** {{Objection}} While Darwin's theory attempts to explain how life evolved, it draws no conclusions at to how it first began. Science has yet to reasonably explain it
**** {{Objection}} Religion is yet to reasonably explain the origins of God. Similar to the chicken and the egg paradox, nothing can be created from nothing
***** {{Objection}} Specifically God of Abraham, Isaac and Jacob has no beginning and no end, He simply has existed forever. For if He was limited by time, then time would be limiting a limitless God
****** {{Objection}} You could apply the same logic to the universe. This is a special pleading case for a God that ignores not only other Gods but also the universe.
****** {{Objection}} You could apply the same logic to the universe. This is a special pleading case for a God that ignores not only other Gods but also the universe.
*** {{Objection}} While Darwin's theory attempts to explain how life evolved, it draws no conclusions at to how it first began. Science has yet to reasonably explain it
* {{Argument for}} Language plays an integral role in the laws of nature and of DNA. As encoded meaning, language is non-material in its ultimate essence. Apart from something akin to the human mind, there are no serious candidates for explaining how linguistic phenomena might otherwise arise. The only reasonable way to account for the linguistic aspects of the laws of nature and of DNA is an intellect with capacities so vast that most people would immediately identify this entity as God.&lt;ref&gt;&quot;A Linguistic Argument for God's existence&quot;, [http://www.etsjets.org/files/JETS-PDFs/58/58-4/JETS_58-4_771-86_Baumgardner&amp;Lyon.pdf Direct Paper link (PDF)]&lt;/ref&gt;
** {{Objection}} The human mind is accounted for more easily by reference to billions of years of biochemical accidents in a massive cosmos. The leap to the “reasonability” of the existence of god is only assumed here, not argued or articulated.
** {{Objection}} This is simply a poor understanding of genetics
*** {{Objection}} This is more of a personal attack than a helpful, edifying argument.
** {{Objection}} Human language plays no role in them at all, integral or otherwise. Genuine laws of nature exist without mind and language, and so does DNA. Laws of nature can be ''expressed in'' language, but do not ''depend on'' language. DNA, a code that gets transcribed into amino acids, ''resembles'' a language, but does not ''depend on'' language and precedes language.
*** {{Objection}} The very fact that it resembles a language makes it count for the original argument
* {{Argument for}} Under a naturalistic worldview, the coincidence of all the cognitive faculties required for knowledge is highly unlikely, since evolution favors survival, not truth, and any overlap of survival and truth is highly implausible.&lt;ref&gt;&quot;IF KNOWLEDGE THEN GOD: THE EPISTEMOLOGICAL THEISTIC ARGUMENTS OF PLANTINGA AND VAN TIL&quot;, [http://www.proginosko.com/docs/If_Knowledge_Then_God.pdf Direct paper link (PDF)]&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.spectator.co.uk/2019/10/donald-hoffmans-the-case-against-reality-is-hard-to-get-your-head-around/|title=Donald Hoffman's The Case Against Reality is hard to get your head around|date=2019-10-12|website=The Spectator|language=en-US}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.quantamagazine.org/the-evolutionary-argument-against-reality-20160421/|title=The Evolutionary Argument Against Reality|last=Amanda Gefter|website=Quanta Magazine|language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://iai.tv/articles/the-evolution-of-reality-auid-1274|title=The Evolution of Reality {{!}} Why the world is not how we see it|date=2019-11-29|website=IAI TV - Changing how the world thinks|language=en-GB}}&lt;/ref&gt; If the reliability of cognitive faculties is low, any belief is not warranted including metaphysical naturalism, which becomes self-defeating.&lt;ref&gt;&quot;Probability And Defeaters&quot;, [http://www.proginosko.com/docs/If_Knowledge_Then_God.pdf Direct Paper link (PDF)]&lt;/ref&gt;
** {{Objection}} Implausible situations become increasingly possible over a cosmic scale — without a god-function.
** {{Objection}} This is simply the God of the Gaps fallacy. Just because something cannot be explained does not mean it is proof of God.
** {{Objection}} This doesn't imply that God exists, only that naturalism is false.
** {{Objection}} The requirement of survival indeed does not require a perfect capacity for truth. However, 1) &quot;any overlap of survival and truth is highly implausible&quot; is clearly untrue: there is a considerable overlap between search for truth and the requirement of survival; and 2) the overlap is far from perfect.
*** {{Objection}} There is often benefit from accurate representation of the world: for instance, when planning the required stock of water to cross a desert, it pays off to make a good estimate of how much is required; and it pays off to avoid a hole instead of falling into it, driving accurate representation of physical environment. Similar requirements impact a huge range of practical human activities.
**** {{Objection}} This reads like an objection to the argument, rather than to the parent objection.
*** {{Objection}} Humans do not have innate intuition for modern physics including general theory of relativity, and yet modern physics seems to approach truth better than human intuition for space. By human intuition, the Earth is still, yet we found that to be untrue.
*** {{Objection}} The human capacity for science is in need of explanation since much of scientific activity is not directly tied to survival of the genes. However, to find the explanation is the task of evolutionary psychology, and finding it may be more challenging than inexpert intuitive reasoning allows.
* {{Argument for}} Chanting of names of God (like Hare Krishna) gives immense happiness to devotees. One can't expect such happiness from a non-theistic worldview, but God existence could explain such happiness.
** {{Objection}} Chanting in unison in a crowd during a sports game can cause similar feelings of happiness. Happy feelings from chanting is not evidence for God.
** {{Objection}} Let us assume that only chanting names of God can give unique exhilarating experience. If that is so, that may well have a Darwinian evolutionary explanation. Even if we do not know for sure what it is, the lack of specific explanation at any point in time does not preclude future discovery of a good explanation. See also [[W:Evolutionary psychology of religion]].
* {{Argument for}} For the Universe to exist, there must be an uncaused cause, God, or the Universe is eternal. So either there's no explanation for God, or there's no explanation for the Universe. The Big Bang is not an explanation, it's a description with no explanation for why it came to be. We then have to rely on chance and happenstance. God fits the picture better.
** {{Objection}} Chance and happenstance are observed but any “causal god” is assumed without presenting evidence or argument.
** {{Objection}} If there must be an uncaused cause, then why can't the Universe be the uncaused cause? Adding God to the chain only adds unnecessary complexity making it a less likely explanation. Just start with a natural, unintelligent and minimally powerful uncaused cause.
*** {{Objection}} Everything that we know in the Universe has a cause external in space and previous in time. Why should the Universe itself be any different? But God is timeless, so the same rules don't necessarily apply.
**** {{Objection}} One could just as easily argue that the Universe is timeless
**** {{Objection}} The Big Bang was the beginning of spacetime. As such, we cannot say that the particles that cause the Big Bang follow the same rules.
** {{Objection}} Not knowing how the Big Bang came about is not proof or evidence that God exists.
*** {{Objection}} While not definitive proof of God, it is definitive proof of the possibility of His Existence.
**** {{Objection}} Something being possible is not evidence that it’s likely, nor a reasonable belief.
** {{Objection}} This assumes the need for an explanation, which is just a [[Wikipedia:False equivalence|false equivalence]]. Things within the Universe require an explanation, but the Universe itself does not require an explanation, because we explain things inside the Universe based on the assumption that there is an external factor already explained. This does not apply to the Universe itself because there is nothing external to the Universe by definition. Either the Universe caused itself or simply has no cause. This might strike many as nonsensical but that is simply because they are unconsciously and inappropriately extending the logic of parts to the whole.
*** {{Objection}} You state the Universe either causes itself or simply had no cause. It's a bald faced assertion with no supporting facts.
*** {{Objection}} You state the Universe either causes itself or simply had no cause. It's a bald faced assertion with no supporting facts.
*** {{Objection}} Everything we know has a cause different from itself. Thus, we must pursue this principle to its natural end, and conclude that the Universe has a cause different from itself, unless contrary evidence can be provided. You can't merely assume that the Universe is all there is and write in an exception into the definition of the Universe.
**** {{Objection}} If you question the origin of the Universe, it's like questioning the water cycle. Where does evaporated water come from? From condensed water. Where does condensed water come from? From precipitated water. Where does precipitated water come from? From ran off water. Where does ran off water come from? From evaporated water. And the cycle goes on and on and on. Similarly, if you question the origin of the Universe, I would say from the Big Bang. If you question the origin of the Big Bang, I would say from the Singularity. If you question the origin of the Singularity, I would say from the Big Crunch. If you question the origin of the Big Crunch, I would say from the Universe. If God does exist, why did he make principles that lead to his non-existence? If God created science, why we can't connect science to him? Even principles are self-looping, so there's also indeed no creator. Energy loops, matter loops, the burnt paper turns into carbon and gas, it just turns into something else to become paper again. The Universe exists in the first place because it loops so there's no beginning or end. The [[Wikipedia:Conservation of energy|law of conservation of energy]] states that energy can neither be created nor destroyed, only be transferred or transformed from one form to another. This implies that energy and the Universe existed in the first place. There's no creator. Just accept the fact that there is no zero in the Universe, there's a fixed number of materials and energy. The Universe is just a continuous loop of energy. It inflates through the Big Bang, reaches the maximum inflation, deflates through the Big Crunch, and reaches the maximum deflation, and the cycle goes on.
***** {{Objection}} I agree; like questioning the water cycle, or our seasons, distinguishing the beginning from the end is a laborious task. However, who's to say in x years from now science won't find evidence reconciling God's existence? If something is currently out of sight and therefore out of mind, does this negate its existence? And emotions: do these not exist because we do not validate them so through our physical senses without having first experienced them? No; emotions make themselves real through their physical manifestations, subject to human interpretation and understood by empathy. Thus, we come to understand one's happiness through its reflexes in the face, voice and body for we have experienced this same sensation ourselves. (Accordingly then, if we have not experienced happiness we would be deficient in empathising and thus recognising happiness). Science seeks to account for the natural phenomena of our world to which we physical beings are inextricably reactive. Yet where technology falls short is in its incompatibility with the human experience; does technology have the same capacity to react impulsively, emotionally and intellectually to its environment? Humans seek personal connections with God, or the divine, as seized by the inarguably human thirst for meaning (again, exempt from the &quot;interests&quot; of insentient technologies). Perhaps science and religion/spirituality should simply be segregated from one another for their languages and interests are incompatible.
* {{Argument for}} God does not have provided conclusive evidence to everyone because not all people want God. 
** {{Objection}} Absence of evidence is not evidence of presence. Absence of evidence is evidence of absence. Lack of evidence doesn't imply that God is fulfilling the desire of people who don't want God, nor that God exists.
*** {{Objection}} Subjective evidence or described subjective experience of God suggest that God can exists who wants to known to people as God choose. And principle of incredulity say that we should believe in experience unless there is good reason to not otherwise. Now if there is positive evidence for non existence of God, it can't counted. But as far as we know, there is no positive evidence of non existence of God. Therefore subjective evidence can't be rejected. Now question one can ask is why would God don't provide subjective evidence to everyone? So, one explanation is not all people want God. 
**** {{Objection}} Subjective evidence for the existence of god is usually sourced in childhood cultural indoctrination and narcissistic wishful thinking.This is not a reliable data set on which to base a fundamental assumption about reality. Subjective evidence like this should be rejected regardless of any lack of proof of god’s non-existence. The lack of objective evidence is for believers to address.
* {{Argument for}} If the Universe is a simulation, then there's a creator beyond spacetime who designed and fine-tuned the simulation. Now either the Universe is a simulation or the Universe is base reality. If the Universe is a simulation, it can be a simulation level 1, level 2, level 1000 or any level. If the Universe is base reality, it can exist in only one way. Therefore, there are more possibilities of the Universe being a simulation than base reality.
** {{Objection}} Even if our Universe is a simulation and has a creator, that doesn't imply a creator of base reality. Our Universe could be a simulation one level below base reality, which implies a creator outside of our local spacetime that exists in base reality, but that does not imply a creator outside spacetime for that base reality since the argument only applies to simulations.
** {{Objection}} There being more possibilities for the simulation hypothesis does not logically necessitate the acceptance of it. The construction violates Occam's razor. If we are allowed to violate Occam's razor and posit an arbitrary number of entities without any evidence, we may 1) posit God, 2) posit God and meta-God who created God, 3) posit also meta-meta-God, 4) posit a chain of meta-Gods of length n, 5) posit an infinite series of Gods, each higher one creating a lower one. All these fanciful hypotheses are unfalsifiable, not bound by observation and experiment, with no basis for differentiating between them. Occam's razor allows us to pick one of the multitude of hypotheses, and do without all these Gods, including meta-God and also God; we can do with the universe itself. Here, the force of Occam's razor was shown without any reference to simulation, but it applies to simulation as well: there being so many constructions of the simulation hypotheses is a weakness, not a strength, of the argument.
* {{Argument for}} Difference between natural and supernatural is artificial. It depends on definition of Supernatural, and gives reason to reject God on artificial grounds. If Supernatural is defined as something which science can't explain, then most of phenomenon can't explainable by science because it depends on our observations, and it doesn't necessarily that observations shows real content of reality out there. If Supernatural is defined as something done that violates laws of physics, there is contradiction in definition. Because laws of physics are not fixed set of laws, but we try to find laws by observations and whatever comes, we describes it as laws of physics. If Supernatural is defined as some actions by agents, it is unclear whether to consider ghost as Supernatural because if some explanation found for it, it becomes natural. And it isn't necessary that nature behaves on it's own. If physical reality doesn't know what it does, or how it does, then it doesn't have to continue to exist. All physical reality can suddenly appears and disappears. It isn't necessarily for reality to such as it should change, it can be such as it remains static for infinity, and nothing happens. It behaves in such a way that is not expected if it run on it's own. However if material nature works under directions of God, it is exactly what we normally expect. And evidence from spontaneous emergent, chaotic unpredictability further suggest material nature can be works under supervision of God, as scripture like Bhagavad Gita also suggest.&lt;ref&gt;{{Cite web|url=https://asitis.com/|title=Bhagavad Gita As It Is Original by Prabhupada|website=asitis.com|access-date=2022-08-16|quote=This material nature is working under My direction, O son of Kunti, and it is producing all moving and unmoving beings. By its rule this manifestation is created and annihilated again and again.}}&lt;/ref&gt;
** {{Objection}} This argument successfully argues god is possible. Few atheists dispute this, though. “Possible” is an extremely low bar for such a significant question.
** {{Objection}} The above looks like an incoherent jumble of multiple arguments. The first step would be to identify the separate arguments and make them dedicated items, and then we could look whether there is something to respond to. As for the definition of &quot;supernatural&quot;, it can be defined by example: a tree spontaneously levitating above the ground with no apparent force to make it so would be a supernatural phenomenon, or would seem to be. Similarly, water spontaneously flowing upwards would be one. Another example is a ghost of a deceased person to haunt a place, perhaps open and close some door. Certain things do not happen in nature. Whether we need to play the genus-differentia game of stating what distinguishes supernatural from natural phenomena is unclear; it may be worthwhile for serious philosophy, but the examples of supernatural phenomena give a good idea. See [[W:Supernatural]] and Hume's [[W:Of Miracles]].
*** {{Objection}} While the initial argument is somewhat incoherent, the objection cites phenomena that were once considered Supernatural and are now explained by Science.  The Truth of God could conceivably follow the path.
* {{Argument for}} The Universe is fine-tuned to support life. This fine-tuning is less surprising and even probable if God exists, but highly unlikely in a godless Universe.&lt;ref&gt;{{Cite journal|last=Chan|first=Man Ho|date=2017-05-24|title=The fine-tuned universe and the existence of God|url=https://repository.hkbu.edu.hk/etd_oa/447|journal=Open Access Theses and Dissertations|quote=To conclude, after a comprehensive study of the fine-tuning arguments, the fine-tuning phenomena strongly support the theistic worldview.}}&lt;/ref&gt;
** {{Objection}} The universe supports life only in certain conditions. Actually, most of it appears extraordinarily hostile to organic chemistry, never mind human life.
** {{Objection}} The universe supports life only in certain conditions. Actually, most of it appears extraordinarily hostile to organic chemistry, never mind human life.
** {{Objection}} If the universe had not been perfectly fine-tuned, we could not have debated this at all. This argument is invalid because it is one of two interpretations of all the same facts; some see fine-tuning as proof of God, others are willing to accept the improbability. If we apply Occam's Razor and the Ultimate Boeing 747 argument, the lack of a creator is the best assumption
** {{Objection}} The puddle analogy refutes the argument of fine tuning. Assume a puddle gains conscience. It might find that the hole it sits in fits it neatly, so neatly in fact that it must have been made for it. The puddle in this scenario does not realize that as it is made of water any hole would fit it perfectly.
*** {{Objection}} The likeliness of a puddle fitting in a hole is nowhere near the extreme unlikeliness of universe existing with all it's features. Hence atomic balance
**** {{Objection}} The likelihood of the universe existing with all its features is 100%.
**** {{Objection}} The likelihood of the universe existing with all its features is 100%.
** {{Objection}} This argument is biased to carbon-based lifeforms. Life could exist in ways that are not based on carbon meaning the fine-tuning of the universe is not as necessary for life to exist as a carbon-based life form might think.
*** {{Objection}} There is no clear evidence of life existing anywhere else in the Universe. Yet, the objection makes the assumption to prove something else doesn't exist
** {{Objection}} Considering multiverse theory, there may be many universes with their own cosmological constants. In this scenario, of course a life form in a &quot;fine tuned&quot; universe sees fine tuning even though that life form's existence can be relegated to chance, not God. This is an example of selection bias. The multitude of other universes inhospitable to life would never develop life forms capable of posing such questions.
* {{Argument for}} If the Universe is ultimately meaningless, devoid of any purpose or design, then all-purpose and meaning one assigns should be imaginary. Because if the Universe doesn't have ultimately any purpose, then life just happened to appear in accidental ways such as it doesn't have any inherent meaning or purpose. However, if God is the reason for the existence of us, there can be the purpose of life. In some religions, it is suggested that God is originally in the spiritual universe with living entities. When a living entity doesn't want God, God makes a material universe for the fulfillment of the desire of them of not wanting God. The material universe is created temporarily such as living entity can realize their connection with God, and when they want God, they can return to God, which may be the purpose of life.
** {{Objection}} This argument assumes its own premise. It relies on the presumption that 'meaning' is an inherent characteristic of life, which is just a rudimentary form of Anthropocentrism. There is no reason why there should be 'meaning' to one's life for life to exist.
*** {{Objection}} While the objection is logical, the ethical ramifications of such thinking lead to nihilism.
**** {{Objection}} Nihilistic or not, a horrible reality must be accepted if it’s a reality. (Besides, meaning can come from creativity and connection.)
**** {{Objection}} Nihilistic or not, a horrible reality must be accepted if it’s a reality. (Besides, meaning can come from creativity and connection.)
**** {{Objection}} The only philosophies we should accept are the logical ones; however, there are secular bases for purpose.
* {{Argument for}} The Universe follows mathematical laws independently of how humans describe them. So mathematics must exist independently of human minds. But all mathematics needs axioms. How can axioms exist independently of human minds? An axiom generator system is needed, or meta-axioms that create the axioms required for mathematical laws. But how can meta-axioms exist? Meta-meta-axioms are needed, and so on. This makes it implausible or even impossible for any mathematical laws to exist. However, it's not impossible if mathematics exists in the mind of God. Because God can conceptualize mathematics.
** {{Objection}} Mathematical laws are how humans describe observed matter and energy. They do not exist independently of us.
** {{Objection}} Axioms are abstract objects that exist in the world of abstract objects, inhabited by numbers, shapes, mathematical functions, mathematical relations, propositions, pure sets, etc. There is no requirement for a mind for abstract objects to exist. If one claims that a mind is needed in order for abstract objects to exist, one does not need to focus on axioms, one may start with numbers as abstract objects that are instantiated or reflected in the empirical universe but are located in the abstract universe. God, being a magician, can explain anything and everything: existence of abstract objects, thunder, disease, floods, trees levitating with no apparent cause, water spontaneously flowing upwards, phenomena observed and those that never occur. The explanatory utility of such an all-explaining entity is in fact zero.
*** {{Objection}} If there is no requirement for a mind for abstract objects to exist, then how did abstract objects come into being? Did they exist eternally before there was mind to perceive them?
*** {{Objection}} If there is no requirement for a mind for abstract objects to exist, then how did abstract objects come into being? Did they exist eternally before there was mind to perceive them?
* {{Argument for}} Objective morality exists and requires an absolute moral authority. Without some absolute authority, all morality is an individual interpretation of morals or shared morality decided by groups of people, which is ultimately subjective. If morality comes just from the survival of fittest, it can be moral to steal or murder, if it results in survival. However, most people regard it as not moral. This absolute authority is equivalent to God who may have created humans and provided some rules or laws which may be inherent in us.
** {{Objection}} The existence of objective morality is not proven. Why it requires an absolute moral “authority” is not explained. Why that authority is god is not explained.
** {{Objection}} The argument assumes its own premise, that objective morality exists, an assumption that isn't necessarily correct but fundamentally necessary for the functionality of the argument. Since, it is from the idea of the existence of objective morality that this argument derives the existence of an absolute moral authority, and from the existence of an absolute moral authority, the existence of God, we necessarily have to conclude that, since no evidence is provided for the positive claim of the existence of objective morality, then no evidence has been provided for the positive claims of the existence of an absolute moral authority or a God. Since the existence of objective morality is provided merely as an assumption lacking supporting evidence, then the existence of an absolute moral authority or the existence of a God is just that, an assumption lacking supporting evidence.
** {{Objection}} Objectively moral facts can exist in a ''world of moral facts'', just like abstract objects like numbers, shapes, pure sets, functions and propositions exist in the world of abstract objects. One only needs to posit such a world, which is what one does if one accepts objectively valid moral facts. The existence of such a world needs as much explanation as the world of abstract objects and the empirical world, meaning none. The requirement that the existence of every single entity (including worlds) must be explained cannot be met: at least, the existence of God is left unexplained. God is just an additional artificial node in the ontology to make sure everything can be bound into a nice tree and each network of nodes stemming from a binary relation has a neat origin node (the cause of all things, the explanation of all phenomena, the first move, the law giver, the source of objective morals, the source of ultimate objectives, etc.; a neat evolutionary cognitive trick, even if incorrect.
* {{Argument for}} Suppose there was no intelligence behind the Universe, no creative mind. In that case, nobody designed my brain for the purpose of thinking. It is merely that when the atoms inside my skull happen, for physical or chemical reasons, to arrange themselves in a certain way, this gives me, as a by-product, the sensation I call thought. But, if so, how can I trust my own thinking to be true? It's like upsetting a milk jug and hoping that the way it splashes itself will give you a map of London. But if I can't trust my own thinking, of course I can't trust the arguments leading to Atheism, and therefore have no reason to be an Atheist, or anything else. Unless I believe in God, I cannot believe in thought: so I can never use thought to disbelieve in God.&lt;ref&gt;The original version of this argument was brought forth by {{W|C. S. Lewis}}.&lt;/ref&gt;
** {{Objection}} A similar argument applies to the theist. Suppose an intelligence designed our brains. This could mean that our brains were designed for thinking rationally, or it could mean that our brains were designed to come to the wrong conclusions. How do we know which is true? We can't. If I can't trust my own thinking, I can't trust the arguments leading to theism. Assuming God exists does not lead to knowledge that we think rationally.
*** {{Objection}} But problem is more serious in universe which just happened to be like it, which is accidental, mindless, purposeless, arbitrary. In such godless universe, there would millions or perhaps billions of coincidence required for brain to function exact right, and there would much more possibility that brain is unreliable. How can anyone know that brain is anything more than quantum fluctuations? In such, how can one can sure that quantum fluctuations should behaves exactly by which it makes logical decision when it doesn't know anything? How can one have faith in such quantum fluctuations such as it shows real reality than shows illusion of reality? However if God creates physical Universe for living entity for giving chance to those who don't want them, then living entity can realize God and can comes to conclusion of God by realizing that material universe is illusionary, and full of suffering which is not ultimate place for him, because of which they can comes to God. In such state, they can have faith in God and can surrender God. Additionally if God wants, God can make known their existence to someone at absolutely certain such as one can becomes certain that God exist, and can realize purpose of existence.
** {{Objection}} Nobody designed the stomach for digesting food either, yet with modern biology, we know through a process of evolution over millions of years that the digestive system evolved naturally through an accumulation of beneficial steps. Just like the eye, just like the mind. In short, evolution provides a better explanation than God.
*** {{Objection}} While Evolution attempts to explain how Life evolved and offers no opinion on how it began.
*** {{Objection}} Not really. How can physical reality behaves so perfectly? If everything is ultimately stochastic thermodynamics progress and quantum fluctuations, why such great coincidence occurs by which it works so well, such as it seems design? If it is not really design but apparent design, it can also be that there is no evolution but apparent evolution, and all entropic thermodynamics progression which doesn't know anything, just happened to behaves like that. But it's seems highly implausible, however it becomes likely if God direct nature or physical reality.
** {{Objection}} If God designed our minds, then why is our reasoning ability so imperfect? Why do people confuse correlation with causation? Why do people believe in astrology and other obvious nonsense?
*** {{Objection}} &quot;Why is our reasoning ability so imperfect?&quot; Generalisation; one person's reasoning ability may seem sufficient in the eyes of one and poor in the other, both judgements relevant to what they simply believe. &quot;Why do people confuse correlation with causation?&quot; Not everything in nature is immediately obvious in its reason, which is why this debate exists in the first place. &quot;Why do people believe in astrology and other obvious nonsense?&quot; Why is this &quot;obviously&quot; nonsense and what are these &quot;others&quot; with which astrology is grouped? Astrology is a science of cause and effect, cause being time and effect the planetary bodies as they interact with the mathematical grid of the skies. Arguably, the effect is further transmitted into our experiences on Earth, the movements themselves thus becoming the authoritative causes in our subordinated lives, though I accept this hierarchy is difficult to prove. In any case, the planets never cease to move, and they do so in cycles, and so they are predictable and patterned both alone as solitary bodies and as a dynamic system; they mirror nature on Earth though they are extraterrestrial phenomena. Do we object to the natural phenomena of Earth that share these qualities: our days and nights and seasons and even our very human selves that run on cycle and patterned behaviour until eventual death? Earth itself is not exempt from this system of planets and very much follow their patterns, and if we strip the specifics from how processes manifest, all phenomena both terrestrial and extraterrestrial are alike. Stripped to the core, that is, all seems to run on the motivation to sustain time's longevity via the cyclical process of regenerated life from decay, or inception from ending, or life from death. Perhaps more bare is the omniscient motive: cause and effect. Denying the effect of astrology then does not mitigate its cause, as per all natural processes scientifically proven to exist or not. If it is the occult sciences to which you object, they are a salve to the bane of mortality yet are scrapped for the very reason that they concern that which can never truly be made known, including the greatest unknown - death - and thus gives the human mind itching for answers great irritability.
*** {{Objection}} Intelligence is a gift endowed to humans that cannot be explained through evolution. It's a gift, but humans are not perfect. If so, we would be divine. How you choose to believe this happened is a matter of faith.{{Clarify}}
* {{Argument for}} If you trust your own thinking, then you must have an absolute perfection, a highest logic, against which to measure your thoughts.
** {{Objection}} Why must that vision of perfection actually exist? Is it not merely a measure in our minds against an imaginary height of perfection? Thought does not imply existence.
** {{Objection}} Untrue: I can trust my own thinking even if it is imperfect. It may be a ''bad'' idea, but I ''can'' do it. The argument has no force.
** {{Objection}} If I do not blindly trust my thinking (as I should not), I should not think in isolation but consider the best arguments the greatest thinkers have to offer. Then I can try to figure out which of the arguments have the greatest force. By giving up the power of my thought entirely, I have no basis for deciding which thinker or which argument to believe. I might just pick what my parents believed, but with a minimum amount of insight I must realize that such a choice is arbitrary and exposes me to whatever they happened to believe, and furthermore, if they used the same reasoning and acquired their belief from their parents, it means that I accept a hereditary theory of knowledge by which no knowledge should ever be acquired and no new arguments discovered since everyone, being properly humble, merely accepts the thought of their ancestors. It seems much more reasonable to combine reasonable doubt about one's reasoning abilities with courage to think and examine arguments and evidence. From which nothing about God follows anyway; what was the argument, exactly? One must not think, therefore, God exists?
* {{Argument for}} A huge number of humans, throughout centuries, have reported all sorts of encounters with God, from the personal internal type to shared apparitions and public miracles. Experiences differ in many ways, but they all support a common cause: the existence of God. It is highly implausible that many reports all be false or misled, and as we trust our experience unless we have good reason to think otherwise, it is reasonable to think that God is reason behind such experiences.
** {{Objection}} Testimonies are generally inconsistent unless they are sufficiently connected by cultural myths or sufficiently vague. Many cultures have the concept of magic, not because magic is real, but because magic is a sufficiently vague concept to hold many different conceptions of it. Unsurprisingly, the reports about God are more similar to the more closely connected the cultures of the witnesses are, which indicates they are cultural phenomena rather than independent observations that corroborate each other.
*** {{Objection}} How testimony is inconsistent? In general experience of God is very difficult to describe or explain. Think how do you describe the experience of the color red to someone who is blind from birth? And cultural explanation applies to everything we experienced. Whatever we experienced is shaped by culture and our background. Like if someone who knows about tree interprets the experience of light for seeing something different than someone who doesn't know about the tree, but it doesn't mean such experience of light is not real. And when we claim that the experience of God must be the same for all, it is not reasonable. Suppose someone prays to God in the form of Lord Krishna, now if God appears as Jesus, he probably requests God to appears in form in which he remembers. Same as if someone remembers God in form of Jesus, and if they experience God in form of other, they may request God that they like God in form of Jesus, therefore, please give me appearance in the form in which I remember. Therefore, it is reasonable to think that God may not experience the same as for every devotee because devotees want God in a specific form they love, and God may fulfill a desire of devotees. Also, it is reasonable to think that God has unlimited forms, and also formless because they are absolute. Also if we reject experience because just they are different, should we also reject the experience of reality if it experienced inconsistently by different people? Most people may answer no. So why should we reject the experience of few billions of people who think they have experienced God? Unless we have positive evidence for the non-existence of God, it shouldn't be rejected.
** {{Objection}} To know that all these testimonies are testimonies about the same thing, we should know their object (God) independently from these testimonies. We should first know the object we are talking about (God) in order to be able to recognize that all reports deal with the same object.
*** {{Objection}} All knowledge ultimately reduces to the testimony of one or more people. Objects cannot be known independently of all testimony.
**** {{Objection}} This claim attempts to homogenize testimony. It fails to recognize not all testimony is created equal; some testimony is backed by evidence while some is not.
** {{Objection}} Eyewitness testimony is notoriously unreliable. Many people have testified seeing thousands of people celebrate the 9/11 attacks in New Jersey, even though this event never took place. If you put an idea in people's minds, some people will believe that they personally saw whatever that idea is, even if the idea turns out to be false.
*** {{Objection}} But people who have experienced God includes medical professionals who know about different psychological phenomenon, and also they have experienced God in which they may differentiate between psychological condition and the real one. Numbers of Medical professionals, physicists, cosmologists experienced God, which makes it less likely to be a purely psychological phenomenon. Additionally, many people go through psychological tests after such experience, in which many are shown to be normal and healthy. If the experience of God is purely delusional, they should show signs of delusion, but instead, people show no such phenomenon. And if God wants, God can make someone know about their existence with absolute certainly.
** {{Objection}} Although there are millions of believers, there are not that many of eyewitnesses, relative to the number of believers.
*** {{Objection}} It can be because the connection with God may be difficult. Some people who love God so much, God may give him direct realization than someone who is not of that level. However many people are spiritually shallow, but they have experienced God slightly like newborn babies experience light slightly and as time passes, he can more clearly experience light. But it doesn't mean most newborn doesn't have any experience of light, and maybe the same as most people who are spiritually shallow, may not experience God deeper.
** {{Objection}} [[Wikipedia:Argumentum ad populum|''Argumentum ad populum'']]''.''
*** {{Objection}} It can argumentum ad populum only if it claims like &quot;People believe in God therefore God exists&quot;. But this is not the same. This argument offers that because millions of people have experienced God, and because we have no positive evidence of the non-existence of God, the experience of God can't be rejected as delusion, hallucinations or lies because else one can reject all experience of all people by same without any positive evidence for its non-existence.
** {{Objection}} Personal experiences can't be accounted for as evidence because there's no evidence to support these reports. How would one prove that these encounters were not a trick of the mind, such as mirages or sleep paralysis, or completely fictional? Experiences of God's existence could easily all be hallucinations, delusions, or attributions of a supernatural cause to natural phenomena which, them being as theists, leading to them being caused by God. I can hallucinate too when I am a theist and I can refer to it as a supernatural occurrence caused by God. It leads to the assumption that it is God-caused because you are a theist.
*** {{Objection}} Personal experience can be accounted for as evidence. Otherwise, it would not be reasonable to believe (unless you personally experience it yourself) that all humans are conscious, certain drugs induce hallucinations or certain psychological phenomena exist, such as dreams, sleep paralysis, Alice in Wonderland syndrome, phantom limb, etc.
*** {{Objection}} When corroborated by so many people, they cannot be so handily dismissed, though.
** {{Objection}} Attributing some encounter with nature or with some unusual phenomenon to the existence of God is a speculative conclusion based on a subjective assessment of the available information. Is contemplation of the beauty of a flower an encounter with God, or simply an appreciation of the fractal nature of the cellular structure that has evolved over millions of years? Are reports of virgin birth evidence of a miracle, or simply a translation error, a misunderstanding of the mechanisms of conception, or marketing hype?
*** {{Objection}} Trying to prove the existence of miracles scientifically is like trying to prove that Gandhi was Indian ''linguistically''. It is the wrong outlet, as we can never re-experience what they did, along with millions of others. These miracles are a matter of faith to them.
* {{Argument for}} Science is built on materialistic assumptions, so it already excludes the existence of God.
** {{Objection}} Current science accepts the existence of many immaterial entities, such as light, energy, spacetime and mathematical entities, so it's not built on materialistic assumptions.&lt;ref&gt;{{Cite journal|last=Moulines|first=C. Ulises|date=1977|title=Por qué no soy materialista|url=http://www.jstor.org/stable/40104059|journal=Crítica: Revista Hispanoamericana de Filosofía|volume=9|issue=26|pages=25–37|issn=0011-1503}}&lt;/ref&gt;
** {{Objection}} Even if science were built on materialistic assumptions and excluded the existence of God, that doesn't imply that God exists.
* {{Argument for}} God is defined as perfect and existence is part of perfection. Therefore, God must exist. See also [[W:Ontological argument]].
** {{Objection}} The concept of &quot;existing smallest positive fraction&quot; contains existence as part of the concept, therefore, the existing smallest positive fraction exists. Makes sense? In the language of philosophers, &quot;existence is not a predicate&quot;&lt;ref&gt;[https://www.philosophyofreligion.uk/theistic-proofs/the-ontological-argument/st-anselms-ontological-argument/existence-is-not-a-predicate/ Existence is not a Predicate], philosophyofreligion.uk&lt;/ref&gt;

=== Con ===

* {{Argument against}} Since there are many religions in the world, all of which have their own idea of God and their own ideas of an afterlife (i.e. heaven/hell, reincarnation, etc.), then which God is real, and which afterlife is real?
** {{Objection}} To claim there is no rational basis for belief in any one religion if at most one can be correct out of thousands is incorrect. The mistake here is assuming all religions have equal probability of being correct, which is not the case: some religions are internally inconsistent, for example, and so have a far lower probability of correctness than more consistent religions. To claim that the credibility of a religion is dictated by its followers' backgrounds is also incorrect; the origin of someone's belief has little bearing on whether the belief is true.
*** {{Objection}} All major religions are internally inconsistent
** {{Objection}} God may have appeared in different parts of the world in different ways so that people of that place and time can understand God, according to circumstances of that time. Therefore, even though God appears different in different religions, it can be same.
*** {{Objection}} If God provides different versions of Himself to different people at different times, then the definition of God is strained. What are we talking about if the definition of God depends on the time and place? Suggesting that God only reveals himself in ways that are not equivalent to His true nature suggests that God hasn't really revealed Himself at all. It suggests he has only revealed caricatures of Himself dependent on the cultures to which he displays these caricatures to.
** {{Objection}} This does not exclude the possibility that one of those religions might turn out to be the correct one, with the correct idea of God and an afterlife, even if that religion contradicts all other religions and this means all religions except for one of them turn out to be wrong.
*** {{Objection}} If at most one religion can be correct, out of the many thousands that exist, and it is possible that they are all wrong, there is no rational basis to believe in any one religion over all the others. Most people who believe in a religion do so because of social reasons, for instance being raised in that religion, or falling in love with and marrying a follower of that religion, rather than any rational basis to believe that their particular religion is any more likely to be true than all the other religions it contradicts. No one religion is obviously superior to all of the others enough to persuade all the followers of the other religions to convert.
**** {{Objection}} We aren't debating over the merits of Christianity or Islam for example--these are matters of faith. We are debating whether a God of some sort exists as a starting ground
***** {{Objection}} It is not about the merits of a religion. This is about a lack of communication (or effective communication) which is weak evidence that there is no God.
**** {{Objection}} All monotheistic religions basically have the same main idea, which is worship of a higher power, a God. The religion itself is a set of values or ideas that one certain group &quot;binds&quot; to a higher power. This may be meant in a way of pleasing or satisfying the higher power, which we as humans often feel the need to do. Take away these values and traditions, which is most likely human made. What we have left is the acknowledgement of a higher power. Monotheistic religions are just different ways of saying the same thing.
***** {{Objection}} Well, you're cherry picking monotheistic religion which already is a set that can include the existence only of a single god, which is then worshiped. If you take a larger set, such as all religion, you get differing beliefs even regarding the nature and even the existence of a higher power. It is not just the traditions that are most likely human made, it is the very notion of god(s), having a clear progression from more utilitarian deities to more abstract ideas which are more resistant to empirical disproof.
* {{Argument against}} God's existence would imply that he can change the past. This would imply that some things happened and didn't happen at the same time and in the same sense. But contradictions are impossible, so not only God doesn't exist: his existence is impossible.
** {{Objection}} Being the perfect father and the most/one of the most selfless beings, God would never decide to change the past or predestine the future, as that would be enslaving all of mankind. If that were to happen, there would be no reason for God to give us the power of free agency in the first place. Therefore, God will not change the past, or predestined our future. He has only foreordained it, meaning, he urges mankind to follow his example (bring yourself to Him, and then others, to avoid hypocrisy), but does not control our lives. With blessings/miracles, he can help, but he does not control our free will.
** {{Objection}} Quantum Entanglement shows contradictions can and do exist, at least as far as humans can currently conceive of them
** {{Objection}} This is not a contradiction. Only the edited version of history actually happened. If God, an omnipotent being, changed the path of history, history is changed. As odd as it seemed, the previous scenario never happened.
*** {{Objection}} However you are still presenting a contradiction: a scenario which existed and never existed.
**** {{Objection}} It might have existed but be wiped out of the history of the universe by built in mechanisms wich are available to change a certain scenario. For example if someone saw something wich they were not supposed to see, God could suddenly hit that person by lightning or induce a stroke wich would cause memory loss and they wouldn’t remember seeing that scenario and God could cut it out or replace it from the history of the universe.
** {{Objection}} That wouldn't be a contradiction, because God would know that He changed the past (per being all-knowing). So things that God changed would have happened at the same time, but not in the same sense. God would be able to distinguish them.
*** {{Objection}} Doing one impossible thing is no more difficult than doing two impossible things.&lt;ref&gt;&quot;For why should God not be able to perform the task in question? To be sure, it is a task—the task of lifting a stone which He cannot lift—whose description is self-contradictory. But if God is supposed capable of performing one task whose description is self-contradictory—that of creating the problematic stone in the first place—why should He not be supposed capable of performing another—that of lifting the stone? After all, is there any greater trick in performing two logically impossible tasks than there is in performing one?&quot; Frankfurt, Harry. &quot;The Logic of Omnipotence&quot; first published in 1964 in ''Philosophical Review'' and now in ''Necessity, Volition, and Love''. Cambridge University Press November 28, 1998 pp.1–2&lt;/ref&gt;
** {{Objection}} God's omnipotence is often described as &quot;can do anything that is not logically impossible&quot;, or similarly defined so as to rule out paradoxes deriving from His omnipotence.
*** {{Objection}} If God's omnipotence is limited by &quot;can do anything that is not logically impossible&quot;, then the fact that God is also defined as omniscient and knowing everything would mean God possesses complete knowledge ahead of time of all things that He will do, and is bound by logic to do what He predicted He would do. This would reduce his omnipotence to complete powerlessness since he would never have any choice at all other than to do what he predicted he would do, and thus he would not have any power at all.
**** {{Objection}} By omnipotence and omniscience, which theists usually refer to, we are speaking of something vastly beyond our understanding and quantifying it in human terms. Omnipotence implies that he has the choice to exercise his omnipotence in any particular way, which he knows what his choice will be. Knowing what you plan to do in a circumstance beforehand by no means makes one powerless
** {{Objection}} If God can change the past, it does not mean he would; this is dictated by his character and motivations, which are particular to different religions. This is similar to the argument that if God can create an immovable object he is not all powerful, but if he cannot then he is not all powerful either: it is wordplay, since such a definition is nonsensical. God is unbounded by the universe's laws.
* {{Argument against}} If we are talking about a God which affects the physical world in some way, then saying that God exists is an empirical statement. But there is no hard evidence supporting that statement, and evidence is necessary to prove an empirical statement.
** {{Objection}} God is outside of the confines of our physical world, and whether or not God is actively involved in this world is not being debated. Therefore, saying that God exists is not an empirical statement, it's a metaphysical statement, and empirical proof for metaphysical statements is not necessary or even possible.
*** {{Objection}} By this logic, there is no such thing as a false statement. Anything can be inferred. I have a 3 headed donkey in my backyard except you can't see it, it doesn't smell, and you can't touch it; there is no physical evidence for its existence whatsoever, but it exists metaphysically, above the plane of our existence. At this point, you have to ask, what do I mean when I say the donkey &quot;exists&quot;? If physical evidence isn't required to say that something exists, then I'm correct when I say that a 3 headed, 5 headed, 50,000 headed, and 50,001 headed donkey all exist in my backyard.
**** {{Objection}} However, donkeys exist as empirical evidence. They are real animals that can be touched or seen. God has never been defined as existing in physical form, e.g, like a man to be seen and touched, the only exception being God the son, Jesus Christ. God is described as the creator of the universe, the origin of all life. Comparing the concept of God to an empirically provable being like a donkey is erroneous. Therefore, there is still room for the existence of false statements, as far as empirically provable things( meaning things that exist in our physical world) exist.
***** {{Objection}} The objection is assuming the donkey is an entity that exists in the metaphysical sense, the same as God.
** {{Objection}} God may have perfect reason for not giving hard evidence to everyone. There are some people who don't like God, don't want God or if they know that God exist, they may becomes envy of God. God may not hides their existence from someone who doesn't want God as fulfilling their desires, and which may be better for them. As scriptures like Bhagavad Gita says, it can be that God manifest in proportion to one surrender to God.&lt;ref&gt;{{Cite web|url=https://asitis.com/|title=Bhagavad Gita As It Is Original by Prabhupada|website=asitis.com|access-date=2022-08-16|quote=Always think of Me and become My devotee. Worship Me and offer your homage unto Me. Thus you will come to Me without fail. I promise you this because you are My very dear friend.}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://asitis.com/|title=Bhagavad Gita As It Is Original by Prabhupada|website=asitis.com|access-date=2022-08-16|quote=Abandon all varieties of religion and just surrender unto Me. I shall deliver you from all sinful reaction. Do not fear.}}&lt;/ref&gt;
* {{Argument against}} God is conceived as all-good, all-knowing and all-powerful. So, if God exists, then under any ordinary definition of evil, evil shouldn't exist. But evil clearly exists. Therefore, (all-good and all-powerful) God does not exist. If there is an all-powerful creator of the world, he or she is not all-good, wishing the best for the creation and the creatures. There is a lot excruciating pain, death and destruction. Human life starts with often painful birth and end with often painful death. Stars and galaxies die, species go extinct, humans die in childhood, old-age diseases develop, humans get murdered, killed in war and tortured, animals get killed by other animals, some animals are kept alive before being fully eaten by their predators, there is a whole predatory network of food chain, etc. See also [[W:Problem of evil]].
** {{Objection}} Good would not be good without evil, just as light wouldn't be light without darkness. It is this precise dichotomy that allows us to distinguish between right and wrong. There would not be a world if good and evil were the same thing, just as if light and darkness were the same thing. A good person is good because they have been confronted to evil, and chose to do the right thing, just like evil people choose evil when they could do good. In more metaphysical terms, it would make more sense to see evil as the absence of good, just like darkness is the absence of light. Light can be studied, darkness cannot. To say that God doesn't exist because evil exists would be akin to saying light cannot exist because darkness exists
** {{Objection}} We do not know what God exactly do if they have perfections. We think God should do this or other, but we can't imagine it from way God would think. We can think if I am photon, I would do this or other, but we don't know what exactly happens when one be a photon. God who is outside of material realism, may have perfect reason to produce seemingly imperfect universe. We don't know God's reason for something, and if we know it we also think that everything is like it should be as best.
** {{Objection}} We define ourselves as moral authority and decides what is good or not, but God may beyond good and evil, the absolute one. Because God is free, if living entity is part and parcel of God, living entity should be free, and because of it they have minutes of free will. Now if living entity not like God or doesn't want God, God may fulfill his desire to becomes independent of God, and creates material world where living entity can be independent of God, and enjoy without God. Now as living entity has free Will, they can choose actions for which they are responsible, to which God doesn't intervene as they respect free will. But if they choose to do something bad, it is entirely on them because they have chosen it from their will. God may created suffering in material world to let someone know that it is not permanent place and living entity can again realize God, and can return to them.
** {{Objection}} God could have given humans the power to do evil. If humans can do evil, then evil can exist despite there being an all-powerful, all-knowing and all-good God. Moral responsibility is not hereditary. If my (grownup) child commits a crime, no society will (or should) blame me for it. Similarly, if a human does evil, we shouldn't blame God for it. Giving the power to do evil is not the same as doing it. God may even be the (metaphysical) cause of evil, while not being morally responsible for it. But no one contends that God raised us that way. He merely gave us the freedom to choose. In fact, because we turned away from his benevolence, evil arose, according to theists. Should we not live with the consequences of our actions and disobedience, then?
*** {{Objection}} This would still make God responsible for evil, albeit indirectly. Moral responsibility is partially hereditary. If I knowingly raise my child in a way that makes it highly likely they will commit a crime when grown up that makes me responsible for that crime and subject to blame. An omniscient God should have known that if he gave humans the freedom to choose, we would do evil. Therefore, he allowed evil to exist, which contradicts the benevolent nature of God.
**** {{Objection}} To understand good, we must understand that it cannot exist without evil. This is how we know what is good and what is not. If good didn't exist, so wouldn't evil. They work together, like light and darkness. To say god is evil because evil exists would be the same as saying light is dark because darkness exists. God is the light in the darkness.
*** {{Objection}} Even if God gave humans free will and the power to do evil, this doesn't imply that there should be evil. A world where free will exists but evil does not is logically possible. God can create any world that is logically possible, so God chose a world where there is unnecessary evil. This contradicts the all-powerful and all-good nature of God.
*** {{Objection}} We haven't dealt with the problem of natural evil. Is cancer also the consequence of human disobedience?
** {{Objection}} Assuming God exists, since he created everything, he also would have created the moral standard. God gave humans free will to commit evil, therefore, if God exists, free will is the moral choice. Humans do not get to say that God is immoral, when God is the one who determines what is and is not moral.
*** {{Objection}} The argument is not that God is &quot;immoral&quot;; it is that God is not benevolent or good-wishing. The argument is that he who can easily intervene to make good but does not is not good-wishing. And he who can create in such a way that only good ensues but does not is not good-wishing either. And he who cannot create in such a way that only good ensues is not all-mighty.
* {{Argument against}} In order to exist, an entity must exist as something. To exist as something, the entity must have positive primary attributes (i.e. I'm a material entity, made up of atoms). All of God's attributes are either negatively defined (ex. omniscience can be reduced to 'without limits of knowledge'), secondary (i.e. good) or relational (i.e. creator). If a god is Creator, then it must be immaterial, as nothing can cause itself. But &quot;immaterial&quot; is a negatively defined term. Therefore a god's substance is undefined. All of this is to say that the god concept is incoherent. If this indeed turns out to be the case, then positive belief in such a concept is not possible.
** {{Objection}} Positive and negative properties are vague notions, often interchangeable. 'Closed' can be reduced to 'not open', just as 'open' can be reduced to 'not closed'. Similarly, 'omniscience' can be reduced to positive terms, like 'with total knowledge' just as it can be reduced to negative ones, like 'without limits of knowledge' or 'without ignorance'. Other properties of God, such as 'all-powerful', can also be thought as either positive or negative: 'with complete power' or 'without limits to its power'.
*** {{Objection}} Even so, saying that something is omniscient is a secondary characteristic - it's telling us what something can do, NOT what it is. If I said humans were an IQ of 120, that doesn't really tell me much of anything about what a human IS (as opposed to saying something like an entity in space/time made up of matter, etc).
**** {{Objection}} Yes, it does. Intelligence is an attribute of humans, is it not?
** {{Objection}} {{W|Dark matter}} and {{W|dark energy}} are entities whose existence is generally accepted by the scientific community, despite the fact that we don't know what they are made of. The fact that we don't know what something is made of doesn't imply that it's made of nothing, or that it doesn't exist.
*** {{Objection}} Dark matter and energy are theoretical.
**** {{Objection}} Global warming, evolution, the Theory of Relativity, and even gravity are also theoretical. This does not mean they are wrong. The scientific community can have almost complete certainty in something but still classify something as &quot;just a theory&quot;.
***** {{Objection}} This is irrelevant to the primary point of the original argument: &quot;All of God's attributes are either negatively defined (ex. omniscience can be reduced to 'without limits of knowledge'), secondary (ex. good) or relational (ex. creator).&quot; The part of the original argument stating that God's substance is undefined is not necessary for the original argument to be correct... if the unnecessary sentence &quot;Therefore a god's substance is undefined.&quot; were left out of the original argument it would be a perfectly valid argument and this objection against it would not work. The main point of the original argument is that in order to exist, that entity must has positive primary attributes, of which there still are none for God. This is a red herring, if we remove that unnecessary sentence from the original argument.
** {{Objection}} How about the fact that the Universe exists in the first place. The fact that the necessary things exist in the first place that leads to the big bang theory and the creation of the Universe. Then the Universe shrinks again into a big matter, crushing everything in its collision, bringing back to the theory of big bang, and the cycle goes on infinite time. What if the Universe is not zero in the first place? The Universe exists without the creation of anything. We can't think of anything that might have created the Universe, because it's just there.
* {{Argument against}} God doesn't exist because of Theophagus, the god-eater. Since Theophagus is god-eating by definition, he has no choice but to eat God. So if God exists, He would immediately cease to exist as a result of being eaten. Unless it's proven that Theophagus doesn't exist, then God doesn't exist.
** {{Objection}} God, being infinite and immaterial, cannot be eaten. If there were somehow a being higher than God, (in this case Theophagus,) then that being would be God.
** {{Objection}} Without any evidence or logical argument for the existence of such a being, there's no reason to believe Theophagus exists.
*** {{Objection}} The same argument against Theophagus works on God: Without any evidence or logical argument for the existence of such a being, there's no reason to believe God exists. So either the argument you raised against Theophagus is valid, in which case it is also valid against God, and thus there is no reason to believe God exists, or the argument you raised against Theophagus is invalid, in which case Theophagus has eaten God and God no longer exists.
**** {{Objection}} If you've read the entire &quot;Arguments for&quot; section, one would see that there are arguments for God's existence
***** {{Objection}} Some Of the arguments you're pointing to are all unsound or plainly invalid. There actually are not better arguments for God's existence than for Theophagus.
** {{Objection}} God is omnipotent and omnipresent, so even if Theophagus exists, God can't be eaten by him.
*** {{Objection}} By its definition, Theophagus eats omnipotent and omnipresent beings.
*** {{Objection}} Nothing about omniscience and omnipotence precludes being eaten.
*** {{Objection}} If God is the most powerful being, and Theophagus can eat God, then Theophagus is more powerful than God, so Theophagus is God, therefore Theophagus/God eats itself and Theophagus/God cease to exist.
** {{Objection}} If Theophagus can eat God, who cannot be eaten, his existence creates a contradiction. Therefore, Theophagus cannot exist
*** {{Objection}} If Theophagus can eat God, who cannot be eaten, then God's existence creates a contradiction. Therefore, God cannot exist. To refute this argument, we must prove that Theophagus does not exist independent of the existence of God.
** {{Objection}} We have two options. Either Theophagus is God or Theophagus is not God. Now attribute of Theophagus is God-like, who need to be omnipotent to eat God, who also need to be omniscient to know everything which requires to eat omnipotent God. If Theophagus is god, he has to eat himself before he eat Actual God. So, if he has eaten himself before eating Actual God, he can't eat God and this all create recursion loop. If Theophagus is not god, he needs to have less than omnipotent and omniscient, by which he can't he eat Actual God, because God is perfect omnipotent who knows everything about Theophagus.
* {{Argument against}} Particles don't have a position until their wave function collapses, and wave functions collapse when observed. From experiments such as the double-slit experiment, we infer that there are uncollapsed wave functions. Therefore, there is no being observing all particles, no omniscient being, no God.
** {{Objection}} The wave function could be an actual manifestation of God that is beyond our comprehension (currently)
** {{Objection}} The wave function could be an actual manifestation of God that is beyond our comprehension (currently)
** {{Objection}} We don't really understand ''how'' observation causes superposition to collapse nor ''how'' a being who is outside of spacetime (or alternately who exists in all of spacetime) would even affect superposition. As {{W|George Berkeley}} argued in his version of {{W|Idealism}}, all of the physical Universe exists ''because'' God is perceiving it.
*** {{Objection}} God of the gaps fallacy.
**** {{Objection}} He was merely pointing out that the original statement need not be always true, nor has any true weight because God exists outside of the physical world.
*** {{Objection}} What we do know about it is that observation causes collapse of the wave function (if you want to claim that being outside space time is somehow different in that regard you'll have to substantiate that claim).
** {{Objection}} The wave function collapse does not happen because of observation per se, but when a wave function interacts with a classical environment. If God is all-powerful, he can observe a wave function without interacting with it.
*** {{Objection}} Being all-powerful is self-contradictory. This is because an all-powerful God would be able to predict the future, but also be able to take actions which would contradict His predictions of the future. Since knowing things is a power, being all-powerful implies being all-knowing. And then, since an all-knowing God would know all of the actions He would take ahead of time, an all-knowing God would know in advance all actions He would ever take, and, in order to prevent any paradoxes and allow God to exist, God's &quot;all-powerful&quot; powers would have to be reduced to just doing what God predicted He would do ahead of time. Thus the whole idea of being all-powerful is nonsense.
**** {{Objection}} This has already been addressed in above arguments, but I'll reiterate--knowing what you will do beforehand does not take away from your freedom of choice. His knowledge is his choice.
***** {{Objection}} Knowing that something will happen effectively means that nothing different will happen. If nothing else will happen then God cannot do something else other than what he foresaw, being then effectively limited in what he can do.
*** {{Objection}} In quantum systems observation is intrinsically linked to the behavior of the system. Your assumption that God is omnipotent is therefore in contradiction which a known feature of the actual world (and being God defined as being omnipotent its very existence is inconsistent with observation of properties of the actual world).
** {{Objection}} If God is all-knowing, he does not need to observe particles to know their position.
*** {{Objection}} When a particle exists in a quantum superposition that can be described using a wave function, prior to wave function collapse, that particle does not actually have any definite position, but just probabilities of being in different locations. This has been experimentally verified in the aforementioned double-slit experiment. So talking about the position of a particle whose wavefunction has not yet collapsed as if it is something definite makes no sense, since such particles can and do exist in multiple locations at the same time, which is what produces the interference fringes in the double-slit experiment, from a particle in different locations interfering with itself in other locations, meaning, it really has no one location. So if God were to perceive such a particle as being at one specific location, God would be incorrect.
* {{Argument against}} Some infinite traits, such as &quot;omniscience&quot;, have a computational complexity equal to infinity, thus the {{W|Kolmogorov complexity}} of a God defined with these attributes is infinite, the prior probability for his existence is epsilon, and &quot;P(X exists) is epsilon&quot; is the statistically literate way of saying &quot;X does not exist&quot;.
** {{Objection}} Pure deterministic processes like computation are not the only means of finding Truth.
* {{Argument against}} Non-theism is the parsimonious worldview.&lt;ref&gt;{{Cite web|url=https://en.wikiversity.org/wiki/Beyond_Theism#Non-Theism_is_the_Null_Hypothesis|title=Beyond Theism - Wikiversity|website=en.wikiversity.org|language=en|access-date=2022-08-16}}&lt;/ref&gt; In other words, following Occam's razor, it avoids assuming existence of entities that add no genuine explanatory value. See [[W:Existence of God#Argument from parsimony]].
** {{Objection}} This isn't so much an argument as it is appeal to authority. And a self-referencing one at that.

== See also ==

* [[Beyond Theism]]
* [[Do humans have free will?]]

== External links ==

* [[Wikipedia:Existence of God]]
* [[Wikipedia:Category:Arguments for the existence of God]]
* [[Wikipedia:Category:Arguments against the existence of God]]
* [https://www.britannica.com/topic/existence-of-God Existence Of God], britannica.com
* [https://www.encyclopedia.com/education/encyclopedias-almanacs-transcripts-and-maps/god-existence God, Existence of], encyclopedia.com
* [https://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/proofs-existence-god Proofs for the Existence of God], encyclopedia.com
* [https://plato.stanford.edu/entries/ontological-arguments/ Ontological Arguments], Stanford Encyclopedia of Philosophy
* [https://www.newadvent.org/cathen/06608b.htm Existence of God], CATHOLIC ENCYCLOPEDIA
* [https://creationwiki.org/Existence_of_God Existence of God], CreationWiki

== Notes and references ==

{{Reflist}}

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This set of pages provides a practical introduction to Receiver Operating Characteristic (ROC) analyses, including sample code and examples generated with multiple popular statistical programs.

== Example ROC Party page ==
*Example [[Evidence based assessment/ROC Party|ROC Party]] page

== ROC sample code ==
*Example [https://docs.google.com/document/d/1Hgl06Oi5p1Yw0UsX7XTbEL4Cw6LnbX5mJ7l88ancdvw/edit ROC sample code] (includes R, SPSS, SAS)

== ROC Paper examples ==
The following papers are a selection that use ROC in their analyses, or illustrate application of DLRs to cases and decision-making, or both.

===In press===
*Cohen, J.R., Adams, Z.W., Menon, S.V., Youngstrom, E.A., Bunnell, B.E., Acierno, R., Ruggiero, K.J., &amp; Danielson, C.K. (in press). How should we assess for depression following a natural disaster? An ROC approach to post-disaster assessments in adolescents and adults. ''Journal of Affective Disorders.''
*Danielson, C.K. Cohen, J., Adams, Z., Youngstrom, E.A., Soltis, K., Amstadter, A., &amp; Ruggiero, K. (in press). Clinical decision making following disasters:  efficient identification of PTSD risk in adolescents. ''Journal of Abnormal Child Psychology.'' 
*Jarrett, M., Van Meter, A.R., Hilton, D., Youngstrom, E.A., &amp; Ollendick, T. (in press). Evidence-based assessment of ADHD in youth using a Receiver Operating Characteristic (ROC) approach. ''Journal of Clinical Child and Adolescent Psychology.'' 
*Jo, B., Findling, R.L., Hastie, T., Youngstrom, E.A., Wang, C.P., Arnold, L.E., Fristad, M.A., Frazier, T.W., Birmaher, B., Gill, M.K., &amp; Horwitz, S.M. (in press). Construction of longitudinal prediction targets using semi-supervised learning. ''Statistical Methods in Medical Research.''
*Mesman, E., Youngstrom, E.A., Juliana, N.K., Nolen, W.A., &amp; Hillegers, M.H.J. (in press). Validation of the Seven Up Seven Down (7U7D) in bipolar offspring: screening and prediction of mood disorders. Findings from the Dutch Bipolar Offspring Study. ''Journal of Affective Disorders.''
*Ong, M. L., Youngstrom, E. A., **Chua, J. J. X., *Halverson, T. F., Horwitz, S. M., Storfer-Isser, A., . . . Findling, R. L. (2016). Comparing the CASI-4R and the PGBI-10M for differentiating bipolar spectrum disorders from other outpatient diagnoses in youth. ''Journal of Abnormal Child Psychology.'' doi:10.1007/s10802-016-0182-4
*Ransom, D.M., Burns, A.R., Youngstrom, E.A., Vaughan, C.G., Sady, M.D., &amp; Gioia, G.A. (in press). Applying an evidence-based assessment model to identify students at risk for perceived academic problems following concussion. ''Journal of the International Neuropsychological Society.'' 
*Salcedo, S., Fristad, M., Birmaher, B., Kowatch, R.A., Gadow, K.D., Phillips, M.L., Frazier, T.W., **Chen, Y-L., Arnold, L.E., Horwitz, S.M., Findling, R.L., &amp; The LAMS Team (in press). Diagnostic efficiency of the Child and Adolescent Symptom Inventory (CASI-4R) depressive subscale for youth mood disorders. ''Journal of Clinical Child and Adolescent Psychology.'' doi: 10.1080/15374416.2017.1280807.
*You, D.S., Youngstrom, E.A., Feeny, N.C., Youngstrom, J.K., &amp; Findling, R.L. (in press). Comparing the diagnostic accuracy of five instruments for detecting posttraumatic stress disorder in youth. ''Journal of Clinical Child and Adolescent Psychology.'' doi:10.1080/15374416.2015.1030754

=== 2016 ===
*Algorta, G. P., Dodd, A. L., Stringaris, A., &amp; Youngstrom, E. A. (2016). Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis. ''European child &amp; adolescent psychiatry'', ''25''(9), 949–957. &lt;nowiki&gt;https://doi.org/10.1007/s00787-015-0815-0&lt;/nowiki&gt;
*Youngstrom, E. A., &amp; Van Meter, A. (2016). Empirically supported assessment of children and adolescents. ''Clinical Psychology: Science and Practice, 23''(4), 327–347. &lt;nowiki&gt;https://doi.org/10.1037/h0101738&lt;/nowiki&gt;
*Frazier TW, Klingemier EW, Beukemann M, Speer L, Markowitz L, Parikh S, Wexberg S, Giuliano K, Schulte E, Delahunty C, Ahuja V, Eng C, Manos MJ, Hardan AY, Youngstrom EA, &amp; Strauss MS (2016). Development of an objective autism risk index using remote eye tracking. ''Journal of the American Academy of Child and Adolescent Psychiatry, 55,'' 301-309.
*Jenkins, M.M., &amp; Youngstrom, E.A. (2016). A randomized controlled trial of cognitive debiasing improves assessment and treatment selection for pediatric bipolar disorder. ''Journal of Consulting and Clinical Psychology, 84,'' 323-333.
*Van Meter, A. R., You, D. S., *Halverson, T., Youngstrom, E. A., Birmaher, B., Fristad, M. A., Kowatch, R.A., Storfer-Isser, A., Horwitz, S.M., Frazier, T.W., Arnold, L.E., Findling, R.L., The LAMS Group. (2016). Diagnostic Efficiency of Caregiver Report on the SCARED for Identifying Youth Anxiety Disorders in Outpatient Settings. ''Journal of Clinical Child &amp; Adolescent Psychology,'' 1-15. doi:10.1080/15374416.2016.1188698

=== 2015 ===
*Youngstrom, E.A., Hameed, A., Mitchell, M., Van Meter, A.R., Freeman, A.J., Perez Algorta, G., White, A.M., Clayton, P., Gelenberg, A., &amp; Meyer, R.E. (2015). Direct comparison of the psychometric properties of multiple interview and patient-rated assessments of suicidal ideation and behavior in an adult psychiatric inpatient sample. ''Journal of Clinical Psychiatry, 76,'' 1676-82. doi:10.4088/JCP.14m09353
*Youngstrom, E.A., &amp; De Los Reyes, A. (2015). Moving towards cost-effectiveness in using psychophysiological measures in clinical assessment: Validity, decision-making, and adding value. ''Journal of Clinical Child and Adolescent Psychology, 44,'' 352-361. doi:10.1080/15374416.2014.913252
*Youngstrom, E.A., *Choukas-Bradley, S., *Calhoun, C.D., &amp; Jensen-Doss, A. (2015). Clinical guide to the evidence-based assessment approach to diagnosis and treatment. ''Cognitive and Behavioral Practice, 22,'' 20-35. doi: 10.1016/j.cbpra.2013.12.005 
*Lindhiem, O., Yu, L., Grasso, D.J., Kolko, D.J., &amp; Youngstrom, E.A. (2015). Adapting the Posterior Probability of Diagnosis (PPOD) index to enhance evidence-based screening: An application to ADHD in primary care. ''Assessment, 22,'' 198-207. doi: 10.1177/1073191114540748
*Prieto, M.L., Youngstrom, E.A., Ozerdem, A., Altinbas, K., Quiroz, D., Aydemir, O., Yalin, N., Geske, J.R., Feeder, S.E., Angst, J., &amp; Frye, M.A. (2015). Different patterns of manic/hypomanic symptoms in depression: A pilot modification of the Hypomania Checklist - 32 to assess mixed depression. ''Journal of Affective Disorders, 172,'' 355-360. doi: 10.1016/j.jad.2014.09.047

=== 2014 ===
*Youngstrom, E.A. (2014). A primer on Receiver Operating Characteristic analysis and diagnostic efficiency statistics for pediatric psychology: We are ready to ROC. ''Journal of Pediatric Psychology, 39,'' 204-221. Special Issue: Quantitative Methodology (Guest Editors: Bryan Karaszia &amp; Kristoffer Berlin); doi: 10.1093/jpepsy/jst062 
*Frazier, T., Youngstrom, E.A., Fristad, M., Demeter, C., Birmaher, B., Kowatch, R., . . . Findling, R. (2014). Improving clinical prediction of bipolar spectrum disorders in youth. ''Journal of Clinical Medicine, 3,'' 218-232. doi: 10.3390/jcm3010218
*Lee, H.J., Joo, Y., Youngstrom, E.A., Yum, S.Y., Findling, R.L., &amp; Kim, H.W. (2014). Diagnostic validity and reliability of a Korean version of the Parent and Adolescent General Behavior Inventories. ''Comprehensive Psychiatry, 55,'' 1730-1737. doi:10.1016/j.comppsych.2014.05.008
*Pendergast, L. L., Youngstrom, E. A., Merkitch, K. G., Moore, K. A., Black, C. L., Abramson, L. Y., &amp; Alloy, L. B. (2014). Differentiating bipolar disorder from unipolar depression and ADHD: the utility of the General Behavior Inventory. ''Psychological Assessment, 26,'' 195-206. doi: 10.1037/a0035138
*Van Meter, A. R., Youngstrom, E. A., Youngstrom, J. K., Ollendick, T., Demeter, C., &amp; Findling, R. L. (2014). Clinical decision-making about child and adolescent anxiety disorders using the Achenbach System of Empirically Based Assessment. ''Journal of Clinical Child &amp; Adolescent Psychology.''

=== 2013 ===
*Youngstrom, E.A. (2013). Future directions in psychological assessment: Combining evidence based medicine innovations with psychology’s historical strengths to enhance utility. ''Journal of Clinical Child and Adolescent Psychology, 42,'' 139-159. doi: 10.1080/15374416.2012.736358.
*Algorta, G.P., Youngstrom, E.A., Phelps, J., *Jenkins, M.M., Youngstrom, J.K., &amp; Findling, R.L. (2013). An inexpensive family index of risk for mood issues improves identification of pediatric bipolar disorder. ''Psychological Assessment, 25,'' 12-22. doi: 10.1037/a0029225

=== 2012 ===
*Freeman, A.J., Youngstrom, E.A., Demeter, C., Youngstrom, J.K., &amp; Findling, R.L. (2012). Portability of a screener for pediatric bipolar disorder to a diverse setting. ''Psychological Assessment, 24,'' 341-351. doi: 10.1037/a0025617 

*Jenkins, M.M., Youngstrom, E.A., Youngstrom, J.K., Feeny, N.C. &amp; Findling, R.L. (2012). Generalizability of evidence-based assessment recommendations for pediatric bipolar disorder. ''Psychological Assessment, 24,'' 269-281. doi: 10.1037/a0025775

=== 2011 ===
*Jenkins, M.M., Youngstrom, E.A., Washburn, J.J., &amp; Youngstrom, J.K. (2011). Evidence-based strategies improve assessment of pediatric bipolar disorder by community practitioners. ''Professional Psychology: Research and Practice, 42,'' 121-129. doi: 10.1037/a0022506

=== 2010 ===
*Horwitz, S.M., Demeter, C., Pagano, M.E., Youngstrom, E.A., Fristad, M.A., Arnold, L.E., Birmaher, B., Gill, M.K., Axelson, D., Kowatch, R.A. &amp; Findling, R.L. (2010). Identification of a childhood bipolar phenotype-the Longitudinal Assessment of Symptoms of Mania Study: Background, design and initial screening. ''Journal of Clinical Psychiatry, 71,'' 1511-1517. doi: 10.4088/JCP.09m05835yel2009
*Youngstrom, E.A., &amp; Kendall, P.C. (2009). Psychological science and bipolar disorder. ''Clinical Psychology: Science and Practice, 16,'' 93-97. doi: 10.1111/j.1468-2850.2009.01149.x 
*Youngstrom, E.A., *Freeman, A.J., &amp; *Jenkins, M.M. (2009). The assessment of bipolar disorder in children and adolescents. ''Child and Adolescent Psychiatric Clinics of North America, 18,'' 353-390. doi: 10.1016/j.chc.2008.12.002
*Meyer, S. E., Carlson, G. A., Youngstrom, E., Ronsaville, D. S., Martinez, P. E., Gold, P. W., Hakak, R., &amp; Radke-Yarrow, M. (2009). Long-term outcomes of youth who manifested the CBCL-Pediatric Bipolar Disorder phenotype during childhood and/or adolescence. ''Journal of Affective Disorders, 113,'' 227-235. doi: 10.1016/j.jad.2008.05.024

=== 2008 ===
*Youngstrom, E.A., *Frazier, T.W., Demeter, C., Calabrese, J.R., &amp; Findling, R.L. (2008). Developing a ten-item mania scale from the Parent General Behavior Inventory for children and adolescents. ''Journal of Clinical Psychiatry, 69,'' 831-839. doi:10.4088/JCP.v69n0517 
*Youngstrom, E.A., *Greene, J., &amp; *Joseph, M. (2008). Comparing the psychometric properties of multiple teacher report instruments as predictors of bipolar disorder in children and adolescents. ''Journal of Clinical Psychology, 64,'' 382-401. doi: 10.1002/jclp.20462

*Henry, D., Pavuluri, M., Youngstrom, E.A., &amp; Birmaher, B. (2008). Accuracy of brief and full forms of a mania rating scale. ''Journal of Clinical Psychology, 64,'' 368-381. doi: 10.1002/jclp.20464
*Meyers, O.I., &amp; Youngstrom, E.A. (2008). A Parent General Behavior Inventory subscale to measure sleep disturbance in pediatric bipolar disorder. ''Journal of Clinical Psychiatry, 69,'' 840-843. doi: 10.1016/j.jad.20009.07.020

=== 2007 ===
*Patel, N.C., Patrick, D.M., Youngstrom, E.A., Strakowski, S.M., &amp; DelBello, M.P. (2007). Response and remission in adolescent mania: Signal detection analyses of the Young Mania Rating Scale. ''Journal of the American Academy of Child and Adolescent Psychiatry, 46,'' 628-635. doi: 10.1097/chi.0b013e3180335ae4 
=== 2006 ===
*Youngstrom, E.A., *Meyers, O.I., Youngstrom, J.K., Calabrese, J.R., &amp; Findling, R.L. (2006). Comparing the effects of sampling designs on the diagnostic accuracy of eight promising screening algorithms for pediatric bipolar disorder. ''Biological Psychiatry, 60,'' 1013-1019. doi: 10.1016/j.biopsych.2006.06.023.

*Frazier, T.W., &amp; Youngstrom, E.A. (2006). Evidence based assessment of attention-deficit/hyperactivity disorder: Using multiple sources of information. ''Journal of the American Academy of Child and Adolescent Psychiatry, 45,'' 614-620. doi: 10.1097/01.chi.0000196597.09103.25

=== 2005 ===

*Youngstrom, E. A., *Meyers, O. I., Demeter, C., Kogos Youngstrom, J., Morello, L., Piiparinen, R., Feeny, N. C., Findling, R. L., &amp; Calabrese, J. R. (2005). Comparing diagnostic checklists for pediatric bipolar disorder in academic and community mental health settings. ''Bipolar Disorders, 7,'' 507-517. doi: j.1399-5618.2005.00269.x 

*Youngstrom, E. A., Findling, R. L., Youngstrom, J. K., &amp; Calabrese, J. R., (2005). Towards an evidence-based assessment of pediatric bipolar disorder. ''Journal of Clinical Child and Adolescent Psychology, 34,'' 433-448. Special Issue: Evidence-Based Assessment. doi: 10.1207/s15374424jccp3403_4
*Youngstrom, E. A., &amp; *Duax, J. (2005). Evidence Based Assessment of Pediatric Bipolar Disorder, Part 1: Base Rate and Family History. ''Journal of the American Academy of Child and Adolescent Psychiatry, 44,'' 712-717. doi: 10.1097/01.chi.0000162581.87710.bd

*Youngstrom, E. A., &amp; Kogos Youngstrom, J. (2005). Evidence Based Assessment of Pediatric Bipolar Disorder, Part 2: Incorporating Information from Behavior Checklists. ''Journal of the American Academy of Child and Adolescent Psychiatry, 44''(8), 823-828. doi: 10.1097/01.chi.0000164589.10200.a4

=== 2004 ===
*Youngstrom, E. A., Findling, R. L., Calabrese, J. R., Gracious, B. L., Demeter, C., DelPorto Bedoya, D., &amp; *Price, M.E. (2004). Comparing the diagnostic accuracy of six potential screening instruments for bipolar disorder in youths aged 5 to 17 years. ''Journal of the American Academy of Child &amp; Adolescent Psychiatry, 43,'' 847-858. doi: 10.1097/01.chi.0000125091.35109.1e

=== 2003 ===
*Danielson, C. K., Youngstrom, E. A., Findling, R. L., &amp; Calabrese, J. R. (2003). Discriminative validity of the General Behavior Inventory using youth report. ''Journal of Abnormal Child Psychology, 31,'' 29-39. doi: 10.1023/A:1021717231272

=== 2002 ===
*Findling, R. L., Youngstrom, E. A., DelPorto, D., Papish-David, R., Townsend, L., *Danielson, C. K., &amp; Calabrese, J. R. (2002). Clinical decision-making using the General Behavior Inventory. ''Bipolar Disorders, 4,'' 34-42. doi: 10.1034/j.1399-5618.2002.40102.

=== 2001 ===
*Youngstrom, E.A., Findling, R.L., *Danielson, C.K., &amp; Calabrese, J.R. (2001). Discriminative validity of parent report of hypomanic and depressive symptoms on the General Behavior Inventory. ''Psychological Assessment, 13,'' 267-276. doi: 10.1037//1040-3590.13.2.267

[[Category:Evidence-based assessment]]</text>
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  <page>
    <title>Net neutrality and 'Restoring Internet freedom'</title>
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        <username>DavidMCEddy</username>
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      <comment>/* U.S. incarceration rate */ why incarceration rates uncorrelated with crime</comment>
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      <text bytes="28598" xml:space="preserve">{{Essay}}

:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].''  

Quite possibly the single most consequential action of the Trump administration short of nuclear war is their efforts to destroy net neutrality.&lt;ref&gt;EFF Comments (2017).  EffectiveDefense.org reply comments (2017).  FCM reply comments (2017).  Wikipedia, &quot;[[w:Net neutrality in the United States|Net neutrality in the United States]],&quot; accessed 2017-09-01&lt;/ref&gt;  They claim they are &quot;restoring Internet freedom&quot;,&lt;ref&gt;FCC Restoring Internet Freedom (2017).&lt;/ref&gt; being the freedom of major Internet access providers like Comcast, Verizon, AT&amp;T and Spectrum (formerly Chartered and TWC) to block, throttle, alter (including stripping encryption), and redirect your requests for information from the Internet.&lt;ref&gt;Engineer's letter (2017).&lt;/ref&gt;

This is an issue that many people have not even heard of.  

It's not well known partly because the mainstream media have a conflict of interest in reporting on it. 

It's consequential, because if Trump's Federal Communications Commission (FCC) succeeds in destroying net neutrality, it will be much harder for individuals and small businesses to reach an audience,&lt;ref&gt;FCM Reply Comments (2017)&lt;/ref&gt; and much harder for Internet entrepreneurs to develop new ways of using the Internet.&lt;ref&gt;Engineer's letter (2017).  EFF Comments (2017).&lt;/ref&gt; That's because Internet access providers have in the past and will in the future, block, throttle, alter, and redirect content they don't like and increase their rates to deliver content at the standard high speeds that everyone now expects.&lt;ref&gt;Engineer's letter (2017).  EFF Comments (2017).&lt;/ref&gt; Internet access providers do not want competition from individuals, small businesses and Internet startups.  

== What is net neutrality, and why is it important? == 

[[w:Net neutrality|Net neutrality]] is the principle that all traffic on the Internet should be treated equally by Internet access providers. 

* ''Net neutrality means that anyone with an Internet connection can compete in the marketplace of ideas based solely on the quality of their presentation.''&lt;ref&gt;{{Citation
| last = Graves | first = Spencer | date = July 17, 2017 
| title = EffectiveDefense.org Comment in Opposition to Restoring Internet Freedom NPRM 
| publisher = U. S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/10717083022106
| accessdate = 2017-08-23}}.  See also EFF Comments (2017, pp. 24ff).&lt;/ref&gt;  

Net neutrality is important, because 

* ''[[Winning the War on Terror|Progress on many and perhaps all substantive issues facing humanity today is blocked, because every countermeasure threatens someone with substantive control over the media]].''&lt;ref&gt;Wikiversity, &quot;[[Winning the War on Terror]]&quot;, accessed 2017-09-02&lt;/ref&gt;   

We next consider a few examples of media bias.  


=== Saudi Arabia and Islamic terrorism === 

[[w:The 28 Pages|A US government document declassified July 15, 2016, included summaries of FBI records from 1999 of incidents apparently funded by the Saudis]] testing US security measures in preparation for the [[w:September 11 attacks|September 11 attacks]].&lt;ref&gt;This included incidents in 1999 where an individual tested security measures on the U.S. Southwest border and two others tried to enter the cockpit of an America West flight.  None of these three seem to have been part of the 19 suicide mass murderers of September 11, 2001.  See [[s:Joint Inquiry into Intelligence Community Activities Before and After the Terrorist Attacks of September 11, 2001/Part 4 (Declassified)]], pp. 418-419 and 433-434. This document is officially available from the web site of the U.S. House Intelligence committee.  However, a link to that web site from the Wikipedia article on &quot;The 28 Pages&quot; is generally unresponsive.  Anyone questioning the veracity of the Wikisource version of this document is encouraged to ask a representative in the U.S. House of Senate about this -- and post any responses to the &quot;Discuss&quot; page associated with this article.&lt;/ref&gt; Moreover, the Bush administration knew this before invading Afghanistan and Iraq.&lt;ref&gt;This information was classified &quot;[[w:Classified information in the United States#Top Secret|Top Secret]]&quot;.  See [[s:Joint Inquiry into Intelligence Community Activities Before and After the Terrorist Attacks of September 11, 2001/Part 4 (Declassified)]].&lt;/ref&gt;

[[Winning the War on Terror|Other documentation establishes that the Saudis have been a primary driver of ISIL.]]&lt;ref&gt;The leadership of ISIL reportedly came, at least initially, almost exclusively from Saddam's military officers:  These men were thrown out of work by the U.S. &quot;[[w:De-Ba'athification|De-Ba'athification]]&quot; program without reasonable alternatives for how to use their time constructively.  The rest of them have largely been inspired by the Wahhabi / Salafist branch of Islam, which has been promoted for decades by Saudi Arabia.  This is the most violent strain of Islam.  Other motivation includes revulsion over the death and destruction created by the U.S.-led invasion of Iraq, the corruption in the post-Saddam Iraqi government sustained by U.S.-imposed censorship of the Iraqi media, and torture of prisoners at places like [[w:Abu Ghraib torture and prisoner abuse|Abu Ghraib]].  For more, see the Wikiversity article on &quot;[[Winning the War on Terror]],&quot; accessed 2017-09-01.  This includes the observation from Lt. Col. Brian Steed, who teaches history at the U.S. Army Command and General Staff College in Leavenworth, is fluent in Arabic and the author of a recent book on ISIS, that islamic terrorists comprise between 0.03 and 0.14 percent of Islam.  This is his personal, professional observation and is not an official policy of the U.S. government.&lt;/ref&gt;   

* ''Why is the US still supporting the Saudis?''

[[File:Ho Chi Minh 1946 and signature (cropped).jpg|thumb|Ho Chi Minh 1946]]

=== Vietnam War ===

Less than three years after Dwight Eisenhower's presidency, he wrote that everyone he had communicated with who was knowledgeable about Vietnam agreed “that had elections been held at the time of the fighting [leading to the defeat of the French in 1954], possibly 80 percent of the population would have voted for the Communist Ho Chi Minh”.&lt;ref&gt;{{Citation
| last = Eisenhower | first = Dwight D. 
| year = 1963 | title = The White House Years: Mandate for change, 1953-1956
| publisher = Doubleday | page = 372}}&lt;/ref&gt; If Eisenhower had supported [[w:Geneva Agreements|Vietnamese elections planned for 1956]], the US would not have supported a government whose mistreatment of its own population was a primary contributor to [[w:Fall of Saigon|its defeat in 1975]].  However, Eisenhower felt unable to do this, apparently because Ho Chi Minh's popularity was virtually unknown in the US.  

*''If all of Eisenhower's sources agreed about this, why was the US public so uninformed?''

Eisenhower was hoping to be elected to a second term as President in 1956 and may not have wanted to explain why he had &quot;lost Vietnam to Communism.&quot;    

[[File:RANDterroristGpsEnd2006.svg|thumb|How terrorist groups end (''n'' = 268): The most common ending for a terrorist group is to convert to nonviolence via negotiations (43 percent), with most of the rest terminated by law enforcement (40 percent).  Groups that were ended by military force constituted only 7 percent.&lt;ref&gt;{{Citation 
| last = Jones 
| first = Seth G. 
| last2 = Libicki 
| first2 = Martin C. 
| author-link = w:Seth Jones
| author2-link = w:Martin C. Libicki
| year = 2008 
| title = How Terrorist Groups End:  Lessons for Countering al Qa’ida
| publisher = RAND Corporation 
| page = 19
| isbn =  978-0-8330-4465-5 
| url = http://www.rand.org/content/dam/rand/pubs/monographs/2008/RAND_MG741-1.pdf
| accessdate = 2015-11-29
}}&lt;/ref&gt;]]

=== How terrorist groups end === 

A 2008 RAND study reported that among the 268 terrorist groups they found that ended between 1968 and 2006, more terrorist groups won than were defeated militarily.  Far more effective were negotiations, like those with the Irish Republican Army in Northern Ireland, and law enforcement.  

*''Why is the West using the least effective approach to terrorism?''

[[File:U.S. incarceration rates 1925 onwards.png|thumb|US incarceration rates 1925-2014]]

=== U.S. incarceration rate ===

After being relatively stable for the 50 years from 1925 to 1975, [[w:United States incarceration rate|the incarceration rate in the US shot up by a factor of five in the last quarter of the twentieth century.]] This increase in incarcerations occurred without a corresponding change in crime rates.  This change has been explained as a product of decisions by mainstream commercial broadcasters to focus on the police blotter while firing nearly all their investigative journalists.  A few popular programs like “60 Minutes” were exceptions.&lt;ref&gt;{{cite book|last1=Sacco|first1=Vincent F|title=When Crime Waves|date=2005|publisher=Sage|isbn=0761927832}}, and {{cite book|last1=Youngblood|first1=Steven|title=Peace Journalism Principles and Practices|date=2017|publisher=Routledge|isbn=978-1-138-12467-7|pages=115–131}}.  See also {{Citation 
| last = Sacco | first = Vincent F. 
| date=May 1995 | title = Media Constructions of Crime 
| journal = Annals of the American Academy of Political &amp; Social Sciences 
| volume = 539 | pages = 141–154}}, reprinted as {{Citation
| last = Sacco | first = Vincent F. 
| editor-last =  Potter | editor-first =  Gary W.
| editor2-last = Kappeler | editor2-first = Victor E.| year = 1998
| title = Constructing Crime: Perspectives on Making News and Social Problems
| publisher = Waveland press | pages = 37-51, esp. 42 
| isbn = 0-88133-984-9}}, and the Wikiversity article on &quot;[[Winning the War on Terror]],&quot; accessed 2017-09-01.&lt;/ref&gt; 

*''The incarceration rate is a function of public perception of crime, which is unrelated to the crime rate,'' at least in the US between 1925 and 2014. 

That works for a couple of reasons. First, the crime rate is low enough that most people's perceptions of crime come primarily from the media. Second, public policy tends to be set by what sounds right, ignoring substantial bodies of research on what policies are actually effective, because they are rarely featured in the mainstream media. 

*''Major broadcasters made out like bandits, while their audiences were largely unaware of what they had lost from the near elimination of investigative journalism.''&lt;ref&gt;On February 29, 2016 [[w:Leslie Moonves|Les Moonves]], President and CEO of [[w:CBS Corporation|CBS]], bragged to an investor conference that the Trump campaign &quot;may not be good for America, but it's damn good for CBS. ... The money's rolling in, this is fun.&quot;  He'd said essentially the same thing the previous December.  And in 2012, Moonves similarly noted that, &quot;Super PACs may be bad for America, but they’re very good for CBS.&quot; {{Citation
| last = Fang | first = Lee | date = February 29, 2016 
| title = CBS CEO: “For Us, Economically, Donald’s Place in This Election Is a Good Thing”
| journal = The Intercept | publisher = First Look Media 
| url = https://theintercept.com/2016/02/29/cbs-donald-trump/
| accessdate = 2017-03-22}}.  See also the discussion of this in the Wikiversity article on &quot;[[Winning the War on Terror]],&quot; accessed 2017-09-01.&lt;/ref&gt;

=== Progress blocked by media bias ===

These examples, and [[Winning the War on Terror|similar analyses of other intractable problems]], can be explained as natural products of two general principles:  

* ''Every media organization in the world sells changes in audience behaviors to the people who pay their bills.''

* ''Media organizations rarely bite the hands that feed them.  They must flinch before disseminating any information that might offend a major advertiser or anyone else with substantive control over their budgets or operations.''&lt;ref&gt;For other examples of problems made intractable by flinching by mainstream media, see the Wikiversity article on &quot;[[Winning the War on Terror]],&quot; accessed 2017-09-01.&lt;/ref&gt;  

Many if not all major problems facing humanity today are impacted by these two issues.  Other factors impact different major problems differently, but media funding and governance is an underappreciated universal issue.&lt;ref&gt;Kahneman (2011).  See also the discussion of human psychology and how people make decisions, based on Kahneman, in {{Citation
| title = Winning the War on Terror 
| publisher = Wikiversity 
| url = https://en.wikiversity.org/wiki/Winning_the_War_on_Terror
| accessdate = 2017-10-16 }}&lt;/ref&gt;

[[File:People's Republic of China Turkey Locator.svg|thumb|Turkey and China are among the leaders if not the leaders in [[w:Censorship of Wikipedia|Censorship of Wikipedia]].]]
Better media in general and net neutrality in particular are threats to major leaders the world over,&lt;ref&gt;[[w:Censorship of Wikipedia|Censorship of Wikipedia]] has been documented in many countries including China, France, Iran, Pakistan, Russia, Saudi Arabia, Syria, Thailand, Tunisia, Turkey, the United Kingdom and Uzbekistan.  However, censorship of Wikipedia in many of these countries has been quite limited, with the well-known exceptions of China and Turkey.&lt;/ref&gt; which explains the “[[w:Great Firewall|Great Firewall]] of China”&lt;ref&gt;{{cite news|last1=Mozur|first1=Paul|title=Baidu and CloudFlare Boost Users Over China’s Great Firewall | url=https://www.nytimes.com/2015/09/14/business/partnership-boosts-users-over-chinas-great-firewall.html | work=The New York Times | date=13 September 2015|language=en}}.  See also [[w:Great Firewall]].&lt;/ref&gt; and why [[w:2017 block of Wikipedia in Turkey|Turkey is blocking Wikipedia]].&lt;ref name=&quot;BBC&quot;&gt;{{Cite news|url=http://www.bbc.com/news/world-europe-39754909|title=Turkish authorities block Wikipedia without giving reason|date=29 April 2017|newspaper=BBC News|accessdate=29 April 2017|language=en-GB}}.  See also [[w:2017 block of Wikipedia in Turkey]].&lt;/ref&gt; Net neutrality makes it easier for the bottom 99.5 percent of the human population to obtain better information on options available to them and to organize to better defend and promote their own interests.&lt;ref&gt;[[File:IncomeInequality9b.svg|thumb|This graph shows the income of the given percentiles plus the average from 1947 to 2010 in 2010 dollars. The 2 columns of numbers in the right margin are the cumulative growth 1970-2010 and the annual growth rate over that period. The vertical scale is logarithmic, which makes constant percentage growth appear as a straight line. From 1947 to 1970, all percentiles grew at essentially the same rate; the light, straight lines for the different percentiles for those years all have the same slope. Since then, there has been substantial divergence, with different percentiles of the income distribution growing at different rates.  For more details see the help file for the &quot;incomeInequality&quot; data in the Ecdat package available from the Comprehensive R Archive Network (CRAN; see r-project.org).]] We say 99.5 percent, because the 99.5th percentile doubled when the average annual income did between 1970 and 2010, but every percentile below that got a smaller share of those productivity improvements:  The incomes of the 99th percentile increased only 73 percent, and the incomes of families making less increased even less.  Meanwhile, the incomes of the top one hundredth of one percent almost tripled. See the accompanying chart.  For more on the relation to income inequality, see EffectiveDefense reply comments (2017)&lt;/ref&gt;

One theory, based on &quot;following the money&quot;, would seem to explain the difficulties in achieving progress in these and other intractable problems:  

* The most important information Americans need to protect their interests '''''is rarely fit to print in the New York Times''','' because it would offend major advertisers -- and is similarly not fit to disseminate in other commercial media.&lt;ref&gt;For a more in-depth discussion of this, see the Wikiversity article on &quot;[[Winning the War on Terror]],&quot; accessed 2107-09-01.&lt;/ref&gt;

[[File:USTelecom1996-2015.svg|thumb|Broadband Capital Expenditures by U.S. Broadband Providers ($ billions, 1996-2015)&lt;ref&gt;{{Citation
| date = December 14, 2016
| title =  Broadband Investment Remains Large, but Ticked Down in 2015
| publisher = USTelecom 
| url = https://www.ustelecom.org/news/press-release/broadband-investment-remains-large-ticked-down-2015
| accessdate = 2017-09-01}}&lt;/ref&gt;]]

== Trump's FCC tries to overturn net neutrality == 

Everyone, including major Internet access providers and the Trump administration officially agrees with net neutrality.&lt;ref&gt;Joint Comments by the American Civil Liberties Union (ACLU) and the Electronic Frontier Foundation (EFF) documented both (a) statements supporting net neutrality and (b) actions inconsistent with those statements by several leading Internet access service providers.  Internet access providers stand to gain billions of dollars in additional revenue by destroying net neutrality;  ACLU and EFF have only their reputations to protect.  ACLU reply comments (2017)&lt;/ref&gt; However, Trump and his supporters claim that the 2015 ''Title II Order'' that made it possible to enforce &quot;net neutrality&quot; increased regulatory uncertainty, which forced major companies in this market to reduce their capital expenditures (CapEx) for investments in new high speed Internet infrastructure.  

The drop in 2015 CapEx sounds big at almost a billion dollars but is less than three quarters of the annual changes since 1996.  This is visible in the accompanying plot of U.S. Broadband CapEx investment, cited but not plotted in the Notice of Proposed Rulemaking (NPRM) on &quot;Restoring Internet Freedom&quot; published May 18 by Trump's FCC.&lt;ref&gt;FCC Restoring Internet freedom&lt;/ref&gt; The damage they claim does NOT make sense if one actually looks at the available data.  

To save the nation from this minuscule damage, which they claim is major, they propose transferring net neutrality enforcement to the US Federal Trade Commission (FTC) and the antitrust division of the Department of Justice.&lt;ref&gt;FCC Restoring Internet Freedom (2017)&lt;/ref&gt; 

Those opposing this action argue the following:  

:1.  History records that the ''Title II Order'' provides the ''only'' way that a typical American will be able to find an Internet access provider who will NOT block, throttle, alter (including stripping encryption), or redirect their requests for information from the Internet.&lt;ref&gt;In addition to the EFF comments (2017), the Engineers' letter (2017), and the Wikipedia article on &quot;[[w:Net neutrality in the United States|Net neutrality in the United States]]&quot;,  Commissioner Clyburn noted in her dissent in FCC Restoring Internet Freedom (2017) that FCC Chairman Pai and others complained in 2015 that the Obama administration had not invested enough time and other resources in economic research to evaluate the impact of their ''Title II Order'' before adopting it.  However, Pai and others proceeded to overturn the ''Title II Order'' without any apparent consideration of research like what they had previously requested.  In fact, there are two economists who are acknowledged leaders in studying &quot;market power&quot;, focusing especially on the structure of telecommunications.  They found, in brief, that there is little honest competition in this market, and this is a problem for the future of the international economy:  Jean Tirole, who won the 2014 Nobel Memorial Prize in Economics for seminal research in this area, and Eli Noam, author and editor of many recent works in this area.  See {{Citation
| last = Crawford | first = Susan P | date = October 14, 2014 
| title = Nobel-Winning Message for the FCC: U.S. telecommunications policy makers long ago turned their backs on Nobel-winning economist Jean Tirole's sensible assessments of private communications utilities -- with disastrous results
| publisher = Bloomberg 
| url = https://www.bloomberg.com/view/articles/2014-10-14/nobel-winning-message-for-the-fcc
| accessdate = June 23, 2017}}, 
{{Citation
| last = Laffont | first = Jean-Jacques  
| last2 = Tirole | first2 = Jean | year = 2000 
| title = Competition in telecommunications 
| publisher = MIT Press 
| isbn = 0-262-12223-5}} and {{Citation
| last = Noam | first = Eli M. | year = 2009 
| title = Media ownership and concentration in America 
| publisher = Oxford U. Pr. 
| isbn = 978-0-19-518852-3}}&lt;/ref&gt; FTC and antitrust enforcement cannot adequately protect consumers, small businesses and Internet startups.&lt;ref&gt;EFF comments (2017)&lt;/ref&gt; [[w:Net neutrality in the United States|Previous abuses by Internet access providers]]&lt;ref&gt;Engineers' letter (2017)&lt;/ref&gt; led to several things:  
::* eight years of increasing activism on this issue, 
::* multiple lesser remedies by the FCC that were blocked by courts, 
::* 3.7 million comments on a proposed FCC action in 2014 
::* that led to the ''Title II Order,'' and 
::* almost 22 million comments on “Restoring Internet freedom” filed by the August 30 deadline.&lt;ref&gt;Per count at [https://www.fcc.gov/ecfs/ fcc.gov/ecfs] with &quot;Proceeding&quot; = &quot;17-108&quot;, checked on 2017-08-30 and 2017-08-31.&lt;/ref&gt;   
&lt;!-- TITLE II ORDER MACROECONOMIC BENEFITS --&gt; 
:2.  All relevant data that is reasonably available and credible indicate that the ''Title II Order'' is working to benefit society as a whole without seriously damaging Internet access providers.&lt;ref&gt;If you know exceptions, please modify this essay appropriately.  Or post links to such data in the &quot;Discuss&quot; page associated with this article.&lt;/ref&gt; The ''New York Times'' said Trump's FCC had to cherry-pick their data to get numbers supporting their desired policy change.&lt;ref&gt;{{Citation
| date = April 29, 2017 
| title = F.C.C. Invokes Internet Freedom While Trying to Kill It
| newspaper = New York Times 
| url = https://www.nytimes.com/2017/04/29/opinion/sunday/fcc-invokes-internet-freedom-while-trying-to-kill-it.html?_r=1
| accessdate = 2017-09-01}}&lt;/ref&gt; Ernesto Falcon, Legislative Council with the Electronic Frontier Foundation, said that the publicly traded companies in that market have not mentioned the ''Title II Order'' in their filings with the Securities and Exchange Commission (SEC), which is the ''only place with credible penalties for misleading comments.''  There, they've said that business is good.&lt;ref&gt;Business could be better for them without the ''Title II Order'', which is why they are fighting it.  For Falcon's comments, see {{Citation
| last = Graves | first = Spencer 
| last2 = Falcon | first2 = Ernesto | date = July 25, 2017 
| title = $15 minimum wage on Aug. 8 ballot in KCMO plus Trump’s attack on net neutrality
| publisher = KKFI 
| url = http://www.kkfi.org/program-episodes/15-minimum-wage-aug-8-ballot-kcmo-plus-trumps-attack-net-neutrality/
| accessdate = 2017-09-01}}.&lt;/ref&gt; [[w:Free Press|Free Press]] analyzed 26 different financial measures that could reflect the impact on the industry of the ''Title II Order''.  Only five of the 26 were negative.  Only two seemed statistically significant, and those showed ''improvements'' (not damage as Trump's FCC claims).  It's not clear if any of these changes resulted from the ''Title II Order''.&lt;ref&gt;We say, &quot;seemed&quot; rather than &quot;were&quot; statistically significant, because the assessment of statistical significance assumes statistical independence, which may not hold.  See the summary in the Appendix to {{Citation
| author = Friends of Community Media | date = August 30, 2017 
| title = Friends of Community Media Reply Comment in Opposition to Restoring Internet Freedom NPRM
| publisher = U.S. Federal Communications Commission 
| pages = 8-11 
| url = https://ecfsapi.fcc.gov/file/1083118363145/FCM%20reply%20comments%20to%20FCC%2017-108.pdf
| accessdate = 2017-08-30}}&lt;/ref&gt;
&lt;!-- CONSUMER BENEFITS --&gt; 
:3.  Since the ''Title II Order,'' consumers have benefitted from being able to use Internet applications and devices that had previously been blocked.&lt;ref&gt;Engineers' letter (2017)&lt;/ref&gt;  
&lt;!-- RESTORING INTERNET FREEDOM WOULD BE DISASTROUNS --&gt;  
:4.  If the FCC decides to move forward with parts of their proposed changes, “the result will have a disastrous effect on innovation in the Internet ecosystem”, according to Internet engineers and pioneers:&lt;ref&gt;Engineers' letter (2017)&lt;/ref&gt; Without enforceable net neutrality, effective deployment of new Internet capabilities would require permissions that would rarely be profitably available to a [[w:Startup company|startup]].  

However, it seems likely that this issue will move from the FCC to Congress, because the stakes are huge, as indicated above.  Ernesto Falcon, Legislative Council for the Electronic Frontier Foundation, said, “People need to take it to the next step, which is ... meet your elected officials, your two Senators and member of the House ..., because it’s only through mobilization that we’ll win this.”&lt;ref&gt;Graves and Falcon (2017)&lt;/ref&gt;

Many consumer advocacy groups are organizing around this issue through a coalition called [https://www.battleforthenet.com/ “BattleForTheNet.com”].

== References == 

* ACLU reply comments (2017) {{Citation
| author = American Civil Liberties Union
|author2=Electronic Frontier Foundation
| date = 2017-08-30
| title = Joint reply comments of the American Civil Liberties Union and the Electronic Frontier Foundation on the Notice of Proposed Rulemaking 
| publisher = U.S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/1083079239110
| accessdate = 2017-09-01}}

* EFF Comments (2017): {{Citation
| date = 2017-07-17 
| title = 
Comments of the Electronic Frontier Foundation on Notice of Proposed Rulemaking
| publisher = U. S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/1083079239110
| accessdate = 2017-08-31}} 

* EffectiveDefense.org reply comments (2017): {{Citation
| last = Graves | first = Spencer | date = 2017-08-30 
| title = EffectiveDefense.org Reply Comments in Opposition to Restoring Internet Freedom NPRM
| publisher = U. S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/10831368603881
| accessdate = 2017-08-31 }}.  See also {{Citation
| title = Winning the War on Terror | publisher = Wikiversity 
| url = https://en.wikiversity.org/wiki/Winning_the_War_on_Terror
| accessdate = 2017-08-31}}.  

* Engineers' letter (2017): {{Citation
| author = &lt;nowiki&gt;191 Internet Engineers, Pioneers, and Technologists&lt;/nowiki&gt;
| date = 2017-07-17 
| title = Joint Comments of Internet Engineers, Pioneers, and Technologists on the Technical Flaws in the FCC’s Notice of Proposed Rule-making and the Need for the Light-Touch, Bright-Line Rules from the Open Internet Order
| publisher = U. S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/1071761547058
| accessdate = 2017-08-31}}

* FCC Restoring Internet Freedom (2017): {{Citation
| author = Federal Communications Commission 
| date = May 18, 2017 
| title = Restoring Internet Freedom, WC Docket 17-108 
| publisher = U.S. Federal Communications Commission 
| url = https://apps.fcc.gov/edocs_public/attachmatch/FCC-17-60A1.pdf
| accessdate = 2017-09-01}}

* FCM Reply Comments (2017): {{Citation
| date = 2017-08-30 
| title = Friends of Community Media Reply Comment in Opposition to Restoring Internet Freedom NPRM
| publisher = U. S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/1083118363145
| accessdate = 2017-08-31}}

* {{Citation | last = Kahneman | first = Daniel | year = 2011
| title = Thinking, Fast and Slow 
| publisher = Farrar, Straus and Giroux
| isbn = 978-0374275631}}

* Wikipedia net neutrality in U.S. (2017): {{Citation
| title = Net neutrality in the United States 
| publisher = Wikipedia 
| url = https://en.wikipedia.org/wiki/Net_neutrality_in_the_United_States
| accessdate = 2017-08-31 }}

== Notes ==
{{reflist}}

[[Category:Original research]]
[[Category:Research]]
[[Category:Political science]]
[[Category:Freedom and abundance]]
[[Category:Telecommunications]]</text>
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      <text bytes="28927" xml:space="preserve">{{Essay}}

:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].''  

Quite possibly the single most consequential action of the Trump administration short of nuclear war is their efforts to destroy net neutrality.&lt;ref&gt;EFF Comments (2017).  EffectiveDefense.org reply comments (2017).  FCM reply comments (2017).  Wikipedia, &quot;[[w:Net neutrality in the United States|Net neutrality in the United States]],&quot; accessed 2017-09-01&lt;/ref&gt;  They claim they are &quot;restoring Internet freedom&quot;,&lt;ref&gt;FCC Restoring Internet Freedom (2017).&lt;/ref&gt; being the freedom of major Internet access providers like Comcast, Verizon, AT&amp;T and Spectrum (formerly Chartered and TWC) to block, throttle, alter (including stripping encryption), and redirect your requests for information from the Internet.&lt;ref&gt;Engineer's letter (2017).&lt;/ref&gt;

This is an issue that many people have not even heard of.  

It's not well known partly because the mainstream media have a conflict of interest in reporting on it. 

It's consequential, because if Trump's Federal Communications Commission (FCC) succeeds in destroying net neutrality, it will be much harder for individuals and small businesses to reach an audience,&lt;ref&gt;FCM Reply Comments (2017)&lt;/ref&gt; and much harder for Internet entrepreneurs to develop new ways of using the Internet.&lt;ref&gt;Engineer's letter (2017).  EFF Comments (2017).&lt;/ref&gt; That's because Internet access providers have in the past and will in the future, block, throttle, alter, and redirect content they don't like and increase their rates to deliver content at the standard high speeds that everyone now expects.&lt;ref&gt;Engineer's letter (2017).  EFF Comments (2017).&lt;/ref&gt; Internet access providers do not want competition from individuals, small businesses and Internet startups.  

== What is net neutrality, and why is it important? == 

[[w:Net neutrality|Net neutrality]] is the principle that all traffic on the Internet should be treated equally by Internet access providers. 

* ''Net neutrality means that anyone with an Internet connection can compete in the marketplace of ideas based solely on the quality of their presentation.''&lt;ref&gt;{{Citation
| last = Graves | first = Spencer | date = July 17, 2017 
| title = EffectiveDefense.org Comment in Opposition to Restoring Internet Freedom NPRM 
| publisher = U. S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/10717083022106
| accessdate = 2017-08-23}}.  See also EFF Comments (2017, pp. 24ff).&lt;/ref&gt;  

Net neutrality is important, because 

* ''[[Winning the War on Terror|Progress on many and perhaps all substantive issues facing humanity today is blocked, because every countermeasure threatens someone with substantive control over the media]].''&lt;ref&gt;Wikiversity, &quot;[[Winning the War on Terror]]&quot;, accessed 2017-09-02&lt;/ref&gt;   

We next consider a few examples of media bias.  


=== Saudi Arabia and Islamic terrorism === 

[[w:The 28 Pages|A US government document declassified July 15, 2016, included summaries of FBI records from 1999 of incidents apparently funded by the Saudis]] testing US security measures in preparation for the [[w:September 11 attacks|September 11 attacks]].&lt;ref&gt;This included incidents in 1999 where an individual tested security measures on the U.S. Southwest border and two others tried to enter the cockpit of an America West flight.  None of these three seem to have been part of the 19 suicide mass murderers of September 11, 2001.  See [[s:Joint Inquiry into Intelligence Community Activities Before and After the Terrorist Attacks of September 11, 2001/Part 4 (Declassified)]], pp. 418-419 and 433-434. This document is officially available from the web site of the U.S. House Intelligence committee.  However, a link to that web site from the Wikipedia article on &quot;The 28 Pages&quot; is generally unresponsive.  Anyone questioning the veracity of the Wikisource version of this document is encouraged to ask a representative in the U.S. House of Senate about this -- and post any responses to the &quot;Discuss&quot; page associated with this article.&lt;/ref&gt; Moreover, the Bush administration knew this before invading Afghanistan and Iraq.&lt;ref&gt;This information was classified &quot;[[w:Classified information in the United States#Top Secret|Top Secret]]&quot;.  See [[s:Joint Inquiry into Intelligence Community Activities Before and After the Terrorist Attacks of September 11, 2001/Part 4 (Declassified)]].&lt;/ref&gt;

[[Winning the War on Terror|Other documentation establishes that the Saudis have been a primary driver of ISIL.]]&lt;ref&gt;The leadership of ISIL reportedly came, at least initially, almost exclusively from Saddam's military officers:  These men were thrown out of work by the U.S. &quot;[[w:De-Ba'athification|De-Ba'athification]]&quot; program without reasonable alternatives for how to use their time constructively.  The rest of them have largely been inspired by the Wahhabi / Salafist branch of Islam, which has been promoted for decades by Saudi Arabia.  This is the most violent strain of Islam.  Other motivation includes revulsion over the death and destruction created by the U.S.-led invasion of Iraq, the corruption in the post-Saddam Iraqi government sustained by U.S.-imposed censorship of the Iraqi media, and torture of prisoners at places like [[w:Abu Ghraib torture and prisoner abuse|Abu Ghraib]].  For more, see the Wikiversity article on &quot;[[Winning the War on Terror]],&quot; accessed 2017-09-01.  This includes the observation from Lt. Col. Brian Steed, who teaches history at the U.S. Army Command and General Staff College in Leavenworth, is fluent in Arabic and the author of a recent book on ISIS, that islamic terrorists comprise between 0.03 and 0.14 percent of Islam.  This is his personal, professional observation and is not an official policy of the U.S. government.&lt;/ref&gt;   

* ''Why is the US still supporting the Saudis?''

[[File:Ho Chi Minh 1946 and signature (cropped).jpg|thumb|Ho Chi Minh 1946]]

=== Vietnam War ===

Less than three years after Dwight Eisenhower's presidency, he wrote that everyone he had communicated with who was knowledgeable about Vietnam agreed “that had elections been held at the time of the fighting [leading to the defeat of the French in 1954], possibly 80 percent of the population would have voted for the Communist Ho Chi Minh”.&lt;ref&gt;{{Citation
| last = Eisenhower | first = Dwight D. 
| year = 1963 | title = The White House Years: Mandate for change, 1953-1956
| publisher = Doubleday | page = 372}}&lt;/ref&gt; If Eisenhower had supported [[w:Geneva Agreements|Vietnamese elections planned for 1956]], the US would not have supported a government whose mistreatment of its own population was a primary contributor to [[w:Fall of Saigon|its defeat in 1975]].  However, Eisenhower felt unable to do this, apparently because Ho Chi Minh's popularity was virtually unknown in the US.  

*''If all of Eisenhower's sources agreed about this, why was the US public so uninformed?''

Eisenhower was hoping to be elected to a second term as President in 1956 and may not have wanted to explain why he had &quot;lost Vietnam to Communism.&quot;    

[[File:RANDterroristGpsEnd2006.svg|thumb|How terrorist groups end (''n'' = 268): The most common ending for a terrorist group is to convert to nonviolence via negotiations (43 percent), with most of the rest terminated by law enforcement (40 percent).  Groups that were ended by military force constituted only 7 percent.&lt;ref&gt;{{Citation 
| last = Jones 
| first = Seth G. 
| last2 = Libicki 
| first2 = Martin C. 
| author-link = w:Seth Jones
| author2-link = w:Martin C. Libicki
| year = 2008 
| title = How Terrorist Groups End:  Lessons for Countering al Qa’ida
| publisher = RAND Corporation 
| page = 19
| isbn =  978-0-8330-4465-5 
| url = http://www.rand.org/content/dam/rand/pubs/monographs/2008/RAND_MG741-1.pdf
| accessdate = 2015-11-29
}}&lt;/ref&gt;]]

=== How terrorist groups end === 

A 2008 RAND study reported that among the 268 terrorist groups they found that ended between 1968 and 2006, more terrorist groups won than were defeated militarily.  Far more effective were negotiations, like those with the Irish Republican Army in Northern Ireland, and law enforcement.  

*''Why is the West using the least effective approach to terrorism?''

[[File:U.S. incarceration rates 1925 onwards.png|thumb|US incarceration rates 1925-2014]]

=== U.S. incarceration rate ===

After being relatively stable for the 50 years from 1925 to 1975, [[w:United States incarceration rate|the incarceration rate in the US shot up by a factor of five in the last quarter of the twentieth century.]] This increase in incarcerations occurred without a corresponding change in crime rates.  This change has been explained as a product of decisions by mainstream commercial broadcasters to focus on the police blotter while firing nearly all their investigative journalists.  A few popular programs like “60 Minutes” were exceptions.&lt;ref&gt;{{cite book|last1=Sacco|first1=Vincent F|title=When Crime Waves|date=2005|publisher=Sage|isbn=0761927832}}, and {{cite book|last1=Youngblood|first1=Steven|title=Peace Journalism Principles and Practices|date=2017|publisher=Routledge|isbn=978-1-138-12467-7|pages=115–131}}.  See also {{Citation 
| last = Sacco | first = Vincent F. 
| date=May 1995 | title = Media Constructions of Crime 
| journal = Annals of the American Academy of Political &amp; Social Sciences 
| volume = 539 | pages = 141–154}}, reprinted as {{Citation
| last = Sacco | first = Vincent F. 
| editor-last =  Potter | editor-first =  Gary W.
| editor2-last = Kappeler | editor2-first = Victor E.| year = 1998
| title = Constructing Crime: Perspectives on Making News and Social Problems
| publisher = Waveland press | pages = 37-51, esp. 42 
| isbn = 0-88133-984-9}}, and the Wikiversity article on &quot;[[Winning the War on Terror]],&quot; accessed 2017-09-01.&lt;/ref&gt; 

*''The incarceration rate is a function of public perception of crime, which is unrelated to the crime rate,'' at least in the US between 1925 and 2014. 

That works for a couple of reasons. First, the crime rate is low enough that most people's perceptions of crime come primarily from the media. Second, public policy tends to be set by what sounds right, ignoring substantial bodies of research on what policies are actually effective, because they are rarely featured in the mainstream media.&lt;ref&gt;Part of this is discussed in &lt;!-- The Heckman Equation--&gt;{{cite Q|Q121010808}}, Wikiversity articles on &quot;[[Social construction of crime and what we can do about it]]&quot; and &quot;[[Research, Education and Economic Growth]]&quot; plus Wikpedia articles on &quot;[[w:Recidivism|recidivism]]&quot; and &quot;[[w:Prisoner reentry|prisoner reentry]]&quot;.&lt;/ref&gt;

*''Major broadcasters made out like bandits, while their audiences were largely unaware of what they had lost from the near elimination of investigative journalism.''&lt;ref&gt;On February 29, 2016 [[w:Leslie Moonves|Les Moonves]], President and CEO of [[w:CBS Corporation|CBS]], bragged to an investor conference that the Trump campaign &quot;may not be good for America, but it's damn good for CBS. ... The money's rolling in, this is fun.&quot;  He'd said essentially the same thing the previous December.  And in 2012, Moonves similarly noted that, &quot;Super PACs may be bad for America, but they’re very good for CBS.&quot; {{Citation
| last = Fang | first = Lee | date = February 29, 2016 
| title = CBS CEO: “For Us, Economically, Donald’s Place in This Election Is a Good Thing”
| journal = The Intercept | publisher = First Look Media 
| url = https://theintercept.com/2016/02/29/cbs-donald-trump/
| accessdate = 2017-03-22}}.  See also the discussion of this in the Wikiversity article on &quot;[[Winning the War on Terror]],&quot; accessed 2017-09-01.&lt;/ref&gt;

=== Progress blocked by media bias ===

These examples, and [[Winning the War on Terror|similar analyses of other intractable problems]], can be explained as natural products of two general principles:  

* ''Every media organization in the world sells changes in audience behaviors to the people who pay their bills.''

* ''Media organizations rarely bite the hands that feed them.  They must flinch before disseminating any information that might offend a major advertiser or anyone else with substantive control over their budgets or operations.''&lt;ref&gt;For other examples of problems made intractable by flinching by mainstream media, see the Wikiversity article on &quot;[[Winning the War on Terror]],&quot; accessed 2017-09-01.&lt;/ref&gt;  

Many if not all major problems facing humanity today are impacted by these two issues.  Other factors impact different major problems differently, but media funding and governance is an underappreciated universal issue.&lt;ref&gt;Kahneman (2011).  See also the discussion of human psychology and how people make decisions, based on Kahneman, in {{Citation
| title = Winning the War on Terror 
| publisher = Wikiversity 
| url = https://en.wikiversity.org/wiki/Winning_the_War_on_Terror
| accessdate = 2017-10-16 }}&lt;/ref&gt;

[[File:People's Republic of China Turkey Locator.svg|thumb|Turkey and China are among the leaders if not the leaders in [[w:Censorship of Wikipedia|Censorship of Wikipedia]].]]
Better media in general and net neutrality in particular are threats to major leaders the world over,&lt;ref&gt;[[w:Censorship of Wikipedia|Censorship of Wikipedia]] has been documented in many countries including China, France, Iran, Pakistan, Russia, Saudi Arabia, Syria, Thailand, Tunisia, Turkey, the United Kingdom and Uzbekistan.  However, censorship of Wikipedia in many of these countries has been quite limited, with the well-known exceptions of China and Turkey.&lt;/ref&gt; which explains the “[[w:Great Firewall|Great Firewall]] of China”&lt;ref&gt;{{cite news|last1=Mozur|first1=Paul|title=Baidu and CloudFlare Boost Users Over China’s Great Firewall | url=https://www.nytimes.com/2015/09/14/business/partnership-boosts-users-over-chinas-great-firewall.html | work=The New York Times | date=13 September 2015|language=en}}.  See also [[w:Great Firewall]].&lt;/ref&gt; and why [[w:2017 block of Wikipedia in Turkey|Turkey is blocking Wikipedia]].&lt;ref name=&quot;BBC&quot;&gt;{{Cite news|url=http://www.bbc.com/news/world-europe-39754909|title=Turkish authorities block Wikipedia without giving reason|date=29 April 2017|newspaper=BBC News|accessdate=29 April 2017|language=en-GB}}.  See also [[w:2017 block of Wikipedia in Turkey]].&lt;/ref&gt; Net neutrality makes it easier for the bottom 99.5 percent of the human population to obtain better information on options available to them and to organize to better defend and promote their own interests.&lt;ref&gt;[[File:IncomeInequality9b.svg|thumb|This graph shows the income of the given percentiles plus the average from 1947 to 2010 in 2010 dollars. The 2 columns of numbers in the right margin are the cumulative growth 1970-2010 and the annual growth rate over that period. The vertical scale is logarithmic, which makes constant percentage growth appear as a straight line. From 1947 to 1970, all percentiles grew at essentially the same rate; the light, straight lines for the different percentiles for those years all have the same slope. Since then, there has been substantial divergence, with different percentiles of the income distribution growing at different rates.  For more details see the help file for the &quot;incomeInequality&quot; data in the Ecdat package available from the Comprehensive R Archive Network (CRAN; see r-project.org).]] We say 99.5 percent, because the 99.5th percentile doubled when the average annual income did between 1970 and 2010, but every percentile below that got a smaller share of those productivity improvements:  The incomes of the 99th percentile increased only 73 percent, and the incomes of families making less increased even less.  Meanwhile, the incomes of the top one hundredth of one percent almost tripled. See the accompanying chart.  For more on the relation to income inequality, see EffectiveDefense reply comments (2017)&lt;/ref&gt;

One theory, based on &quot;following the money&quot;, would seem to explain the difficulties in achieving progress in these and other intractable problems:  

* The most important information Americans need to protect their interests '''''is rarely fit to print in the New York Times''','' because it would offend major advertisers -- and is similarly not fit to disseminate in other commercial media.&lt;ref&gt;For a more in-depth discussion of this, see the Wikiversity article on &quot;[[Winning the War on Terror]],&quot; accessed 2107-09-01.&lt;/ref&gt;

[[File:USTelecom1996-2015.svg|thumb|Broadband Capital Expenditures by U.S. Broadband Providers ($ billions, 1996-2015)&lt;ref&gt;{{Citation
| date = December 14, 2016
| title =  Broadband Investment Remains Large, but Ticked Down in 2015
| publisher = USTelecom 
| url = https://www.ustelecom.org/news/press-release/broadband-investment-remains-large-ticked-down-2015
| accessdate = 2017-09-01}}&lt;/ref&gt;]]

== Trump's FCC tries to overturn net neutrality == 

Everyone, including major Internet access providers and the Trump administration officially agrees with net neutrality.&lt;ref&gt;Joint Comments by the American Civil Liberties Union (ACLU) and the Electronic Frontier Foundation (EFF) documented both (a) statements supporting net neutrality and (b) actions inconsistent with those statements by several leading Internet access service providers.  Internet access providers stand to gain billions of dollars in additional revenue by destroying net neutrality;  ACLU and EFF have only their reputations to protect.  ACLU reply comments (2017)&lt;/ref&gt; However, Trump and his supporters claim that the 2015 ''Title II Order'' that made it possible to enforce &quot;net neutrality&quot; increased regulatory uncertainty, which forced major companies in this market to reduce their capital expenditures (CapEx) for investments in new high speed Internet infrastructure.  

The drop in 2015 CapEx sounds big at almost a billion dollars but is less than three quarters of the annual changes since 1996.  This is visible in the accompanying plot of U.S. Broadband CapEx investment, cited but not plotted in the Notice of Proposed Rulemaking (NPRM) on &quot;Restoring Internet Freedom&quot; published May 18 by Trump's FCC.&lt;ref&gt;FCC Restoring Internet freedom&lt;/ref&gt; The damage they claim does NOT make sense if one actually looks at the available data.  

To save the nation from this minuscule damage, which they claim is major, they propose transferring net neutrality enforcement to the US Federal Trade Commission (FTC) and the antitrust division of the Department of Justice.&lt;ref&gt;FCC Restoring Internet Freedom (2017)&lt;/ref&gt; 

Those opposing this action argue the following:  

:1.  History records that the ''Title II Order'' provides the ''only'' way that a typical American will be able to find an Internet access provider who will NOT block, throttle, alter (including stripping encryption), or redirect their requests for information from the Internet.&lt;ref&gt;In addition to the EFF comments (2017), the Engineers' letter (2017), and the Wikipedia article on &quot;[[w:Net neutrality in the United States|Net neutrality in the United States]]&quot;,  Commissioner Clyburn noted in her dissent in FCC Restoring Internet Freedom (2017) that FCC Chairman Pai and others complained in 2015 that the Obama administration had not invested enough time and other resources in economic research to evaluate the impact of their ''Title II Order'' before adopting it.  However, Pai and others proceeded to overturn the ''Title II Order'' without any apparent consideration of research like what they had previously requested.  In fact, there are two economists who are acknowledged leaders in studying &quot;market power&quot;, focusing especially on the structure of telecommunications.  They found, in brief, that there is little honest competition in this market, and this is a problem for the future of the international economy:  Jean Tirole, who won the 2014 Nobel Memorial Prize in Economics for seminal research in this area, and Eli Noam, author and editor of many recent works in this area.  See {{Citation
| last = Crawford | first = Susan P | date = October 14, 2014 
| title = Nobel-Winning Message for the FCC: U.S. telecommunications policy makers long ago turned their backs on Nobel-winning economist Jean Tirole's sensible assessments of private communications utilities -- with disastrous results
| publisher = Bloomberg 
| url = https://www.bloomberg.com/view/articles/2014-10-14/nobel-winning-message-for-the-fcc
| accessdate = June 23, 2017}}, 
{{Citation
| last = Laffont | first = Jean-Jacques  
| last2 = Tirole | first2 = Jean | year = 2000 
| title = Competition in telecommunications 
| publisher = MIT Press 
| isbn = 0-262-12223-5}} and {{Citation
| last = Noam | first = Eli M. | year = 2009 
| title = Media ownership and concentration in America 
| publisher = Oxford U. Pr. 
| isbn = 978-0-19-518852-3}}&lt;/ref&gt; FTC and antitrust enforcement cannot adequately protect consumers, small businesses and Internet startups.&lt;ref&gt;EFF comments (2017)&lt;/ref&gt; [[w:Net neutrality in the United States|Previous abuses by Internet access providers]]&lt;ref&gt;Engineers' letter (2017)&lt;/ref&gt; led to several things:  
::* eight years of increasing activism on this issue, 
::* multiple lesser remedies by the FCC that were blocked by courts, 
::* 3.7 million comments on a proposed FCC action in 2014 
::* that led to the ''Title II Order,'' and 
::* almost 22 million comments on “Restoring Internet freedom” filed by the August 30 deadline.&lt;ref&gt;Per count at [https://www.fcc.gov/ecfs/ fcc.gov/ecfs] with &quot;Proceeding&quot; = &quot;17-108&quot;, checked on 2017-08-30 and 2017-08-31.&lt;/ref&gt;   
&lt;!-- TITLE II ORDER MACROECONOMIC BENEFITS --&gt; 
:2.  All relevant data that is reasonably available and credible indicate that the ''Title II Order'' is working to benefit society as a whole without seriously damaging Internet access providers.&lt;ref&gt;If you know exceptions, please modify this essay appropriately.  Or post links to such data in the &quot;Discuss&quot; page associated with this article.&lt;/ref&gt; The ''New York Times'' said Trump's FCC had to cherry-pick their data to get numbers supporting their desired policy change.&lt;ref&gt;{{Citation
| date = April 29, 2017 
| title = F.C.C. Invokes Internet Freedom While Trying to Kill It
| newspaper = New York Times 
| url = https://www.nytimes.com/2017/04/29/opinion/sunday/fcc-invokes-internet-freedom-while-trying-to-kill-it.html?_r=1
| accessdate = 2017-09-01}}&lt;/ref&gt; Ernesto Falcon, Legislative Council with the Electronic Frontier Foundation, said that the publicly traded companies in that market have not mentioned the ''Title II Order'' in their filings with the Securities and Exchange Commission (SEC), which is the ''only place with credible penalties for misleading comments.''  There, they've said that business is good.&lt;ref&gt;Business could be better for them without the ''Title II Order'', which is why they are fighting it.  For Falcon's comments, see {{Citation
| last = Graves | first = Spencer 
| last2 = Falcon | first2 = Ernesto | date = July 25, 2017 
| title = $15 minimum wage on Aug. 8 ballot in KCMO plus Trump’s attack on net neutrality
| publisher = KKFI 
| url = http://www.kkfi.org/program-episodes/15-minimum-wage-aug-8-ballot-kcmo-plus-trumps-attack-net-neutrality/
| accessdate = 2017-09-01}}.&lt;/ref&gt; [[w:Free Press|Free Press]] analyzed 26 different financial measures that could reflect the impact on the industry of the ''Title II Order''.  Only five of the 26 were negative.  Only two seemed statistically significant, and those showed ''improvements'' (not damage as Trump's FCC claims).  It's not clear if any of these changes resulted from the ''Title II Order''.&lt;ref&gt;We say, &quot;seemed&quot; rather than &quot;were&quot; statistically significant, because the assessment of statistical significance assumes statistical independence, which may not hold.  See the summary in the Appendix to {{Citation
| author = Friends of Community Media | date = August 30, 2017 
| title = Friends of Community Media Reply Comment in Opposition to Restoring Internet Freedom NPRM
| publisher = U.S. Federal Communications Commission 
| pages = 8-11 
| url = https://ecfsapi.fcc.gov/file/1083118363145/FCM%20reply%20comments%20to%20FCC%2017-108.pdf
| accessdate = 2017-08-30}}&lt;/ref&gt;
&lt;!-- CONSUMER BENEFITS --&gt; 
:3.  Since the ''Title II Order,'' consumers have benefitted from being able to use Internet applications and devices that had previously been blocked.&lt;ref&gt;Engineers' letter (2017)&lt;/ref&gt;  
&lt;!-- RESTORING INTERNET FREEDOM WOULD BE DISASTROUNS --&gt;  
:4.  If the FCC decides to move forward with parts of their proposed changes, “the result will have a disastrous effect on innovation in the Internet ecosystem”, according to Internet engineers and pioneers:&lt;ref&gt;Engineers' letter (2017)&lt;/ref&gt; Without enforceable net neutrality, effective deployment of new Internet capabilities would require permissions that would rarely be profitably available to a [[w:Startup company|startup]].  

However, it seems likely that this issue will move from the FCC to Congress, because the stakes are huge, as indicated above.  Ernesto Falcon, Legislative Council for the Electronic Frontier Foundation, said, “People need to take it to the next step, which is ... meet your elected officials, your two Senators and member of the House ..., because it’s only through mobilization that we’ll win this.”&lt;ref&gt;Graves and Falcon (2017)&lt;/ref&gt;

Many consumer advocacy groups are organizing around this issue through a coalition called [https://www.battleforthenet.com/ “BattleForTheNet.com”].

== References == 

* ACLU reply comments (2017) {{Citation
| author = American Civil Liberties Union
|author2=Electronic Frontier Foundation
| date = 2017-08-30
| title = Joint reply comments of the American Civil Liberties Union and the Electronic Frontier Foundation on the Notice of Proposed Rulemaking 
| publisher = U.S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/1083079239110
| accessdate = 2017-09-01}}

* EFF Comments (2017): {{Citation
| date = 2017-07-17 
| title = 
Comments of the Electronic Frontier Foundation on Notice of Proposed Rulemaking
| publisher = U. S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/1083079239110
| accessdate = 2017-08-31}} 

* EffectiveDefense.org reply comments (2017): {{Citation
| last = Graves | first = Spencer | date = 2017-08-30 
| title = EffectiveDefense.org Reply Comments in Opposition to Restoring Internet Freedom NPRM
| publisher = U. S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/10831368603881
| accessdate = 2017-08-31 }}.  See also {{Citation
| title = Winning the War on Terror | publisher = Wikiversity 
| url = https://en.wikiversity.org/wiki/Winning_the_War_on_Terror
| accessdate = 2017-08-31}}.  

* Engineers' letter (2017): {{Citation
| author = &lt;nowiki&gt;191 Internet Engineers, Pioneers, and Technologists&lt;/nowiki&gt;
| date = 2017-07-17 
| title = Joint Comments of Internet Engineers, Pioneers, and Technologists on the Technical Flaws in the FCC’s Notice of Proposed Rule-making and the Need for the Light-Touch, Bright-Line Rules from the Open Internet Order
| publisher = U. S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/1071761547058
| accessdate = 2017-08-31}}

* FCC Restoring Internet Freedom (2017): {{Citation
| author = Federal Communications Commission 
| date = May 18, 2017 
| title = Restoring Internet Freedom, WC Docket 17-108 
| publisher = U.S. Federal Communications Commission 
| url = https://apps.fcc.gov/edocs_public/attachmatch/FCC-17-60A1.pdf
| accessdate = 2017-09-01}}

* FCM Reply Comments (2017): {{Citation
| date = 2017-08-30 
| title = Friends of Community Media Reply Comment in Opposition to Restoring Internet Freedom NPRM
| publisher = U. S. Federal Communications Commission 
| url = https://www.fcc.gov/ecfs/filing/1083118363145
| accessdate = 2017-08-31}}

* {{Citation | last = Kahneman | first = Daniel | year = 2011
| title = Thinking, Fast and Slow 
| publisher = Farrar, Straus and Giroux
| isbn = 978-0374275631}}

* Wikipedia net neutrality in U.S. (2017): {{Citation
| title = Net neutrality in the United States 
| publisher = Wikipedia 
| url = https://en.wikipedia.org/wiki/Net_neutrality_in_the_United_States
| accessdate = 2017-08-31 }}

== Notes ==
{{reflist}}

[[Category:Original research]]
[[Category:Research]]
[[Category:Political science]]
[[Category:Freedom and abundance]]
[[Category:Telecommunications]]</text>
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      <text bytes="14994" xml:space="preserve">—Knowing what to do
== Introduction ==
[[File:Compass rose browns 00.png|thumb|right| 250px|[[w:Moral_reasoning|Moral Reasoning]] is the thought process we go through to determine what we ought to do. ]]
{{TOC right | limit|limit=2}}
[[w:Moral_reasoning|Moral Reasoning]] is the thought process we go through to determine what we ought to do. Moral reasoning helps us decide what is right and what is wrong. 

No simple rule, list of commandments, formula, or outcome seems to adequately capture the complexities of moral reasoning.&lt;ref&gt;The two major approaches to moral reasoning are [[w:Deontological_ethics|deontological]]—establishing rules, and [[w:Consequentialism|consequentialism]]—favoring certain outcomes. Each approach taken alone has its strengths and weaknesses. &lt;/ref&gt;  Instead, we need to rely on a toolkit that taken together provides guidance in a variety of situations.

This course presents the elements of the moral reasoning toolkit.
== Objectives ==
{{100%done}}{{By|lbeaumont}}
The objective of this course is to improve your moral reasoning and your moral behavior.

All students are welcome and there are no prerequisites to this course.
 
The course contains many [[w:Hyperlink|hyperlinks]] to further information. Use your judgment and these [[What Matters/link following guidelines|link following guidelines]] to decide when to follow a link, and when to skip over it.

A PowerPoint [https://archive.org/details/moral-reasoning presentation based on this course] is available at the Internet Archive.

This course is part of the [[Wisdom/Curriculum|Applied Wisdom curriculum]] and of the [[Virtues/Moral_Reasoning|Moral Reasoning curriculum]].

If you wish to contact the instructor, please [[Special:EmailUser/Lbeaumont|click here to send me an email]] or leave a comment or question on the [[Talk: Moral Reasoning|discussion page]].

Students may benefit by choosing a specific [[Moral_Reasoning/Moral_Issues|moral issue]] to focus on throughout the course. Choose an issue from this [[Moral_Reasoning/Moral_Issues|list of moral issues]] or any other source to focus on as you study and reflect on each lesson.

Elements of moral reasoning include the topics addressed in each of the following sections.

== A Basis for Moral Reasoning ==
[[w:Moral_realism|Moral realists]] seek an objective basis for morality. So far none has been found. Philosopher [[w:David_Hume|David Hume]] is often credited with stating that you cannot [[w:Is–ought_problem | derive ''ought'' from ''is'']], leading many to conclude there can be no objective basis for moral reasoning. 

Three major approaches to moral reasoning are:
#[[w:Deontology|Deontology]]—morality is best expressed as a set of rules,
#[[w:Consequentialism|Consequentialism]]—morality is assessed by the consequences of actions, and
#[[w:Virtue_ethics|Virtue ethics]]—the concept of moral virtue is central to ethics.

Unfortunately, simple examples can show these approaches lead to inconsistent results.

However, [[w:Steven_Pinker|Steven Pinker]] assures us “when you combine [[w:Self-interest|self-interest]] and [[w:Sociality|sociality]] with ''[[w:Impartiality|impartiality]]''—the interchangeability of perspectives—you get the core of morality.”&lt;ref&gt;{{cite book |last=Pinker |first= Steven |author-link=w:Steven_Pinker|date= September 28, 2021 |title=Rationality: What It Is, Why It Seems Scarce, Why It Matters  |url= |publisher= Viking |pages=432 |isbn= 978-0525561996 }} @ 106 of 608.&lt;/ref&gt;

This course adopts that assumption, and the following sections apply the principle of impartiality to several specific situations.

== Moral Virtue ==
Moral reasoning begins with the [[w:Virtue|moral virtues]].

[[Virtues|Moral virtue]] is excellence in being for the good.
=== Assignment ===
#Complete the Wikiversity course on [[Virtues|Moral Virtue]].
#Become excellent in being for the good.
#Complete the Wikiversity course on [[Clarifying values]].
#Use your well-chosen values to guide your actions. 

== Intellectual Honesty ==
[[Intellectual honesty]] combines communicating in [[Virtues/Good Faith|good faith]] with a primary motivation toward [[Seeking True Beliefs|seeking true beliefs]].
=== Assignment ===
#Complete the Wikiversity course on [[Intellectual honesty]].
#Become Intellectually  honest.
#Insist on Intellectual honesty from others. 
== Fairness ==
[[Understanding Fairness|Fairness]] is freedom from bias, dishonesty, or injustice.

We naturally appeal to fairness to avoid or resolve conflict. Unfortunately, when conflict emerges it is often difficult for adversaries to agree on what is actually fair. 
=== Assignment ===
#Complete the Wikiversity course on [[Understanding Fairness]].
#Seek fairness.
#Treat people fairly.
== Global Perspective ==
When we adopt a [[Global Perspective|Global perspective]] we consider all that matters.
=== Assignment ===
#Complete the Wikiversity course on [[Global Perspective]].
#Adopt a global perspective when seeking, identifying, defining, and solving problems.
#Complete the Wikiversity course on [[Grand challenges]].
#Use the grand challenges to establish priorities. 
== Flourishing ==
[[File:Double-alaskan-rainbow.jpg|thumb|right|300px|Flourish! ]]
Promote human [[Flourishing|flourishing]]—where people often have positive experiences.

We flourish when we focus on [[What Matters|what matters]].
=== Assignment ===
#Read the on-line book [[Motivation and emotion/Book/2018/Flourishing|''Flourishing: What is flourishing and how can it happen?'']]
#Complete the Wikiversity course on [[What Matters]].
#Focus on What Matters.
#Don’t be distracted by things that don’t matter.

== Live the Golden Rule ==
We [[Living the Golden Rule|live the golden rule]] when we treat others only as we consent to being treated in the same situation.

We can test our moral reasoning by ''changing places'' with the people most affected by our moral decisions and evaluating their experience.
=== Assignment ===
#Complete the Wikiversity course on [[Living the Golden Rule]].
#Treat others only as you consent to being treated in the same situation.
#Live the Golden Rule.
== Respect Human Dignity ==
Moral reasoning requires us to always respect [[Dignity|human dignity]]—the quality of worth and honor intrinsic to every person.
=== Assignment ===
#Complete the Wikiversity course on [[Dignity]].
#Treat every person with dignity.
#Insist on being treated with dignity
== Protect Human Rights ==
Moral reasoning requires us to protect [[w:Human_rights|human rights]], worldwide.

Human rights provide essential protections for every person.
=== Assignment ===
#Complete the Wikiversity course on [[Assessing Human Rights]].
#Read the essay [[Assessing Human Rights/Beyond Olympic Gold|Beyond Olympic Gold]]. 
#Work to protect human rights, worldwide.
== Face Facts ==
[[Facing Facts/Reality is our common ground|Reality is our common ground]]. Face facts, especially when they are difficult.
=== Assignment ===
#Complete the Wikiversity course on [[Facing Facts]].
#Embrace reality. Draw on reality when [[Finding Common Ground|seeking common ground]].
#[[Facing Facts|Face facts]], especially when they are difficult.
#Do not argue matters of fact, [[Thinking Scientifically|research them]] using [[w:Wikipedia:Reliable_sources|reliable sources]] and reliable methods.
#[[Knowing How You Know|Know how you know]].
#Read the essay [[Exploring Worldviews/Aligning worldviews|aligning worldviews]].
#Align your worldview with reality.
#Complete the course on [[Finding Common Ground|finding common ground]].
#Find common ground.

== True Beliefs ==
Untrue beliefs are more likely to be [[w:Harm|harmful]] than true beliefs.&lt;ref&gt;Many ill-fated undertakings were based on untrue beliefs. These [[Grand_challenges#Case_Studies|case studies]] provide several examples.&lt;/ref&gt; Therefore, we have a moral obligation to choose true beliefs.&lt;ref&gt;{{Cite web|url=https://philosophyasawayoflife.medium.com/the-ethics-of-belief-f1d459c572e3|title=The ethics (or lack thereof) of belief|last=Life|first=Philosophy as a Way of|date=2022-08-31|website=Medium|language=en|access-date=2022-09-06}}&lt;/ref&gt;
=== Assignment ===
#Complete the Wikiversity course on [[Seeking True Beliefs]].
#Choose true beliefs.
#Challenge untrue beliefs.
#[[Living_Wisely/advance_no_falsehoods|Advance no falsehoods]].

== Transcend Conflict ==
Whenever you encounter conflict, work to transcend it.
=== Assignment ===
#Complete the Wikiversity course on [[Transcending Conflict]].
#When you encounter conflict, work to transcend it.
#Complete the course on [[Finding Common Ground|finding common ground]].
#Find common ground.

== Find Courage ==
[[w:Jane_Addams|Jane Addams]] tells us that &quot;Action indeed is the sole medium of expression for ethics.&quot;

Find the courage to act according to your well-chosen values.
=== Assignment ===
#Complete the Wikiversity course on [[Finding Courage]].
#Find your courage.
#Use your courage to act on your well-chosen [[w:Value_(ethics)|values]].
== Resolve Anger ==
Resolve anger without resorting to violence.
=== Assignment ===
#Complete the Wikiversity course on [[Resolving Anger]].
#Resolve anger without resorting to violence.

== Scope ==
An important moral question is who matters? Possible answers include: me, my friends and family, my community, my [[w:tribe|tribe]], my nation, all the world’s people, [[w:Sentience|sentient]] beings, all beings, all living species, people yet to be born, all beings for all time.

[[q:Albert_Einstein#1950s|Albert Einstein said]]: &quot;A human being is a part of the whole, called by us 'Universe', a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the rest — a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole nature in its beauty. Nobody is able to achieve this completely, but the striving for such achievement is in itself a part of the liberation and a foundation for inner security.&quot;

The scope of moral reasoning extends to include all sentient beings, worldwide, now and into the future.

[[Living Wisely/Seeking Real Good|Seek real good]]. [[Doing Good|Do good]].

== Moral Development ==
Continue to [[w:Moral_development|develop your moral reasoning]]. Continue to refine, adapt, and evolve your moral reasoning to respond more adequately to increasingly difficult and complex moral issues. Consider the various [[w:Lawrence_Kohlberg's_stages_of_moral_development#Formal_elements|formal elements of moral development]] and work to advance to higher levels of moral reasoning. 

== Applying Moral Reasoning ==
Moral reasoning is useful when it helps us analyze and decide moral issues fairly and determine what we ought to do.
=== Assignment ===
#Choose a moral issue to analyze from this list of [[/Moral Issues/]], or from another source.
#Apply your moral reasoning to decide the moral issue.
#What did you decide you ought to do?
#Continue to apply your moral reasoning to make careful decisions throughout your life.
#Continue to refine your moral reasoning.

== Further Reading ==
Students interested in learning more about moral reasoning may be interested in the following materials:
*{{Cite book|title=The bonobo and the atheist: in search of humanism among the primates|last=Waal|first=Frans B. M. de|date=2013|publisher=W.W. Norton &amp; Company|isbn=978-0-393-07377-5|location=New York, NY}}
*{{Cite book|title=The origins of virtue: human instincts and the evolution of cooperation|date=1998|publisher=Penguin Books|isbn=978-0-14-026445-6|editor-last=Ridley|editor-first=Matt|series=A Penguin book : science|location=London}}
*{{cite book |last=Gensler |first=Harry J. |date=March 21, 2013 |title=Ethics and the Golden Rule |publisher=Routledge |pages=256 |isbn=978-0415806879}}
*{{cite book |last=Greene |first=Joshua |date=December 30, 2014 |title=Moral Tribes: Emotion, Reason, and the Gap Between Us and Them |publisher=Penguin Books |pages=432 |isbn=978-0143126058 }}
* {{cite book |last=Hicks |first=Donna |date=January 29, 2013 |title=Dignity: Its Essential Role in Resolving Conflict |publisher=Yale University Press |pages=240 |isbn=978-0300188059}}
* {{cite book |last=Rosling |first=Hans |date=April 3, 2018 |title=Factfulness: Ten Reasons We're Wrong About the World--and Why Things Are Better Than You Think |url= |publisher=Flatiron Books |pages=352 |isbn=978-1250107817 |author-link=w:Hans_Rosling }}
*{{cite book |last=Frankfurt |first=Harry G. |date=October 31, 2006 |title=[[w:On_Truth|On Truth]] |publisher=Knopf |pages=112 |isbn=978-0307264220 |author-link=w:Harry_Frankfurt }}
* {{cite book |title=Global Perspectives: A Handbook for Understanding Global Issues |last1=Kelleher |first1=Ann |last2=Klein |first2=Laura |year=2005 |publisher=Prentice Hall |isbn=978-0131892606 |pages=240}}
* [[Motivation and emotion/Book/2018/Flourishing|Flourishing: What is flourishing and how can it happen?]]
*{{cite book |last=Seligman |first=Martin E. P.  |date=February 7, 2012 |title=Flourish: A Visionary New Understanding of Happiness and Well-being  |publisher=Atria Books |pages=368 |isbn=978-1439190760 |author-link=w:Martin_Seligman }}
*{{cite book |last=Reich |first=Robert B. |date=February 20, 2018 |title=The Common Good   |publisher=Knopf |pages=208 |isbn=978-0525520498 |author-link=w:Robert_Reich }}
*{{cite book |last=Kidder |first=Rushworth M. |date=April 1, 1994 |title=Shared Values for a Troubled World: Conversations with Men and Women of Conscience  |publisher=Jossey-Bass |pages=332 |isbn=978-1555426033  }}
*{{cite book |last=Christakis |first=Nicholas A. |date=March 26, 2019 |title=Blueprint: The Evolutionary Origins of a Good Society |publisher=Little, Brown Spark |pages=544 |isbn=978-0316230032 |author-link=w:Nicholas_A._Christakis}}
*{{cite book |last=Pinker |first= Steven |author-link=w:Steven_Pinker|date= September 28, 2021  |title=[[w:Rationality_(book)| Rationality: What It Is, Why It Seems Scarce, Why It Matters]]| publisher= Viking |pages=432 |isbn= 978-0525561996 }}
* Books from the reading list at: http://www.librarything.com/list/10594/all/Secular-Ethics 
* Consider the ethics courses at: http://cohe.humanistinstitute.org/?page_id=9646
I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research.

* {{cite book|title=Ethics in the Real World: 90 Essays on Things That Matter|last=Singer|first=Peter|date=April 18, 2023|publisher=Princeton University Press|isbn=978-0691237862|pages=488|author-link=w:Peter_Singer}}

== References ==
&lt;references/&gt;
{{Moral Reasoning}}
{{CourseCat}} 
[[Category:Life skills]]
[[Category:Applied Wisdom]]
[[Category:Philosophy]]
[[Category:Courses]]
[[Category:Ethics]]</text>
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      <text bytes="3096" xml:space="preserve">Many innovations can be inspired and imagined by reversing the viewpoint, components, users, and other elements of an existing product or solution. Consider how the various reversals listed here might transform an old idea into a new one, identify an opportunity, or offer a new viewpoint that can help reframe a problem. 
[[File:Apollo 10 earthrise.jpg|thumb| Our thinking changes when we reverse our point of view.]]
{{div col}}
Cause ↔ Effect

Customer ↔ Supplier

Bottom up ↔ Top down

Emergent ↔ Designed

[[Exploring Social Constructs|Social constructs]] ↔ Brute facts

Distributed ↔ Centralized

Giver ↔ Recipient

Scarce ↔ Abundant

Awful ↔ Awesome

Payer ↔ Collector

Share ↔ Hoard

Talk ↔ Listen

Input ↔ Output

In ↔ Out

Top ↔ Bottom

Upright ↔ Sideways

Left ↔ Right

Big ↔ Small

Close ↔ Distant

Near ↔ Far

Fast ↔ Slow

Strong ↔ Weak

Forward ↔ Backward

Movable ↔ Fixed

Dynamic ↔ Static

Linear ↔ Circular

Linear ↔ Radial

Circular ↔ Spiral

Spiral ↔ Helix

Helix ↔ Möbius strip

Microscope ↔ Telescope

Focused ↔ Circumspect

Complex ↔ Simple

Black ↔ White

Objective ↔ Subjective

Light ↔ Dark

Day ↔ Night

Light ↔ Heavy

Foreground ↔ Background

Forest ↔ Trees

Headline ↔ Footnote

Push ↔ Pull

Assignment ↔ Invitation

Fall ↔ Rise

Obvious ↔ Cryptic

Obvious ↔ Incorrect

Known ↔ Unknown

Master ↔ Slave

Powerful ↔ Powerless

Own ↔ Rent

Democracy ↔ Tyranny

Comedy ↔ Tragedy

Play ↔ Work

Teacher ↔ Student

Teach ↔ Learn

Doctor ↔ Patient

Expert ↔ Novice

Predator ↔ Prey

Victim ↔ Perpetrator

Guilty ↔ Innocent

Male ↔ Female

Rich ↔ Poor

Growing ↔ Shrinking

Fearful ↔ Confident

Timid ↔ Bold

Costly ↔ Free

Constrained ↔ Free

Reader ↔ Writer

Problem ↔ Solution

Good ↔ Bad

Good ↔ Evil

Certain ↔ Unlikely

Rich ↔ Poor

Rough ↔ Smooth

Soft ↔ Hard

Sharp ↔ Round

Edgy ↔ Cozy

Frightening ↔ Comforting

Maximize ↔ Minimize

Negative ↔ Positive

Cost ↔ Benefits

Difficult ↔ Easy

Active ↔ Passive

Accuser ↔ Accused

Competition ↔ Cooperation

Win Win ↔ Zero Sum

Not ↔ Now

Specialize ↔ Generalize

Temporary ↔ Permanent

Aggregate ↔ Separate

Isolate ↔ Congregate

Divide ↔ Unite

Component ↔ System

Minority ↔ Majority

Old ↔ Young

Generous ↔ Greedy

Secret ↔ Transparent

Copyright ↔ Copy left

Kind ↔ Cruel

Helpful ↔ Hurtful

Humble ↔ Arrogant

Vindictive ↔ Compassionate 

Authority ↔ Expertise

Loyalty ↔ Fairness

Me ↔ You

Ideological ↔ Empirical

Theory ↔ Practice

Rigid ↔ Flexible

Near ↔ Far

Fast ↔ Slow

Fear ↔ Curiosity

Local ↔ Global

Tribal ↔ Global

War ↔ Peace

Love ↔ Hate

Reification ↔ Instantiation

Symptom ↔ Cause

Power ↔ Reason

Passion ↔ Reason

Anecdote ↔ Representative evidence

Despair ↔ Hope

Advocate ↔ Criticize

Personal ↔ Communal

Strange ↔ Charm

Tuba ↔ Sousaphone

Left-handed ↔ Right-handed

Noun ↔ Verb

Verb ↔ Preposition

{{div col end}}

{{CourseCat}}</text>
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      <comment>/* Confidence limits */ improve label on Fig 7</comment>
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      <text bytes="69459" xml:space="preserve">{{Research project}} 
:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]], and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].''  

This article (i) describes efforts to model the time between the first test of a nuclear weapon by one nation and the next over the 74 years of history since the first such test by the US,&lt;ref&gt;This is being written on 2020-04-26.  For the purposes of the present analysis, this is considered to be 74 years since the first test of a nuclear weapon on 1945-07-16.&lt;/ref&gt; (ii) forecasts nuclear proliferation over the next 74 years with statistical error bounds quantifying the uncertainty, and (iii) reviews some of the geopolitical questions raised by this effort.  Our modeling effort considers the possibility that the rate of nuclear proliferation may have slowed over time.   

In brief, current international policy seems to imply that nuclear proliferation can be ignored.  The analysis in this article of the statistical and non-statistical evidence suggests that nuclear proliferation is likely to continue unless (a) a nuclear war destroys everyone's ability to make more such weapons for a long time, or (b) an international movement has far more success than similar previous efforts in providing effective nonviolent recourse  for grievances of the poor, weak and disfranchised.  

Statistical details are provided in R Markdown vignettes on “Forecasting nuclear proliferation” and &quot;GDPs of nuclear weapon states&quot; in an appendix, below.  Those vignettes should allow anyone capable of accessing the {{w|free and open-source software}} [[R (programming language)|R]] and [[w:RStudio|RStudio]] to replicate this analysis and modify it in any way they please to check the robustness of the conclusions. 

== The data ==

The “nuclearWeaponStates” dataset&lt;ref&gt;{{cite Q|Q88894684}}&lt;!-- nuclearWeaponStates dataset--&gt;&lt;/ref&gt; in the Ecdat package for R&lt;ref&gt;{{cite Q|Q56452356}}&lt;!-- https://github.com/sbgraves237/Ecdat --&gt;&lt;/ref&gt; was used for this study.  Those data combine information from the “World Nuclear Weapon Stockpile” maintained by Ploughtshares,&lt;ref&gt;{{cite Q|Q63197617}}&lt;!--  World Nuclear Weapon Stockpile compiled by Ploughshares --&gt;&lt;/ref&gt; the Wikipedia article on “[[w:List of states with nuclear weapons|List of states with nuclear weapons]]”, and multiple articles in the {{w|Bulletin of the Atomic Scientists}}.  This includes the five states that officially had nuclear weapons when the United Nations {{w|Treaty on the Non-Proliferation of Nuclear Weapons}} (Non-Proliferation Treaty, NPT) entered into force in 1970 (the US, Russia, the UK, France and China) plus four others that first tested nuclear weapons since (India, Israel, Pakistan, and North Korea).  

There seems to be a fairly broad consensus on the dates of the first tests of 8 of these 9 nuclear weapon states.  Some reports claim that France and Israel had such close collaboration on nuclear weapons development in the late 1950s that the first test of a nuclear weapon by France on 1960-02-13 effectively created two nuclear-weapon states, not one.&lt;ref&gt;{{cite Q|Q88922617}}&lt;!-- The Third Temple's Holy of Holies:  Israel's Nuclear Weapons, tech report by Lt.Col. Warner D. Farr, --&gt;&lt;/ref&gt; The current study used the date of the 1979-09-22 {{w|Vela Incident}} for Israel.  A 2019 report by Professor Avner Cohen, professor at the Middlebury Institute of International Studies, and the Director of the Education Program and Senior Fellow at the James Martin Center for Nonproliferation Studies, said that, “there is a scientific and historical consensus that [the Vela incident] was a nuclear test and that it had to be Israeli”,&lt;ref&gt;{{cite Q|Q88921529}}&lt;!--  U.S. Covered Up an Israeli Nuclear Test in 1979, Foreign Policy Says, article in Haaretz --&gt;&lt;/ref&gt; conducted probably with South Africa.  A robustness analysis could involve simply deleting Israel as a separate nuclear-weapon state.

== Plotting the time between the “first test” by one nuclear-weapon state and the next == 

[[File:NucWeaponStates YrsBetw1stTsts.svg|thumb|Figure 1.  Years between new nuclear-weapon states.   CN = China, FR = France, GB = UK, IL = Israel, IN = India, KP = North Korea, PK = Pakistan, RU = Russia.  NPT = {{w|Treaty on the Non-Proliferation of Nuclear Weapons}} (Non-Proliferation Treaty).  INF = {{w|Intermediate-Range Nuclear Forces Treaty}}.  The US is not on this plot, because it had no predecessors.]]

A plot of times between &quot;first tests&quot; by the world's nuclear-weapon states as of 2020-04-29 suggests that the process of nuclear proliferation has slowed;  see Figure 1.  

This plot also marks the effective dates of both the {{w|Treaty on the Non-Proliferation of Nuclear Weapons}} (Non-Prolireration Treaty, NPT) and the [[w:Intermediate-range Nuclear Forces Treaty|Intermediate-range Nuclear Forces (INF) Treaty]] (1970-03-05 and 1988-06-01, respectively), because of the suggestion that those treaties may have slowed the rate of nuclear proliferation.

A visual analysis of this plot suggests that nuclear proliferation is still alive and well, and neither the NPT nor the INF treaty impacted nuclear proliferation. The image is pretty bad:  There were only 5 nuclear-weapon states when the NPT entered into force in 1970.&lt;ref&gt;{{cite Q|Q91335914}}&lt;!-- Treaty on the Non-Proliferation of Nuclear Weapons --&gt;.  See also {{w|reaty on the Non-Proliferation of Nuclear Weapons}}.&lt;/ref&gt; When US President {{w|George W. Bush}} decried an [[w:Axis of evil|&quot;Axis of evil&quot;]] in his State of the Union message, 2002-01-29,&lt;ref&gt;{{cite Q|Q91337578}}&lt;!--2002 State of the Union Address by US President George W. Bush--&gt;.  See also [[w:Axis of Evil]].&lt;/ref&gt; there were 8.  As this is written 2020-04-21, there are 9.  

Toon et al. (2007) noted that in 2003 another 32 had sufficient fissile material to make nuclear weapons if they wished.  Moreover, those 32 do ''NOT'' include either Turkey nor Saudi Arabia.  On 2019-09-04, Turkish President Erdogan said it was unacceptable for nuclear-armed states to forbid Turkey from acquiring its own nuclear weapons.&lt;ref&gt;{{cite Q|Q91338524}}&lt;!-- Erdogan says it's unacceptable that Turkey can't have nuclear weapons, 2019 Reuters news article by Ece Toksabay--&gt;;  {{cite Q|Q91342138}} &lt;!-- Tom OConnor (2019) “Turkey has U.S. nuclear weapons, Now it says it should be allowed to have some of its own” --&gt;.&lt;/ref&gt;

Similarly, in 2006 ''Forbes'' reported that Saudi Arabia has &quot;a secret underground city and dozens of underground silos for&quot; Pakistani nuclear weapons and missiles.&lt;ref&gt;{{cite Q|Q91342270}}&lt;!-- Forbes:2006:  AFX News Limited: &quot;Saudia Arabia working on secret nuclear program with Pakistan help - report&quot; --&gt;;  see also [[w:Nuclear program of Saudi Arabia]].&lt;/ref&gt; In 2018 the ''Middle East Monitor'' reported that &quot;Israel 'is selling nuclear information' to Saudi Arabia&quot;.&lt;ref&gt;{{cite Q|Q91343477}}&lt;!-- Israel ‘is selling nuclear information’ to Saudi Arabia, per Middle East Monitor --&gt;;  see also [[w:Nuclear program of Saudi Arabia]].&lt;/ref&gt; This is particularly disturbing, because of the substantial evidence that Saudi Arabia may have been and may still be the primary recruiter and funder of Islamic terrorism.&lt;ref&gt;{{cite Q|Q55616039}}&lt;!-- Medea Benjamin (2016) Kingdom of the Unjust: Behind the US-Saudi Connection --&gt;; see also [[Winning the War on Terror]].&lt;/ref&gt;

This analysis suggests that the number of nuclear-weapon states will likely continue to grow until some dramatic break with the past makes further nuclear proliferation either effectively impossible or sufficiently undesirable. 

This article first reviews the data and history on this issue. We then discuss modeling these data as a series of annual Poisson observations of the number of states conducting a first test of a nuclear weapon each year (1 in each of 8 years since 1945; 0 in the others).

A relatively simple model for the inhomogeneity visible in Figure 1 is {{w|Poisson regression}} assuming that log(Poisson mean) is linear in the time since the first test of a nuclear weapon by the US on 1945-07-16.&lt;ref&gt;A vignette on “Forecasting nuclear proliferation” describes fitting such models to the available data in a way that allows anyone able to run the {{w|free and open-source software}} {{w|R (programming language)}} to [[w:Reproducibility#Reproducible research|reproduce the analysis outlined in this article]] and experiment with alternatives:  {{cite Q|Q89780728}}&lt;!-- Forecasting nuclear proliferation--&gt;.&lt;/ref&gt;  This model is plausible to the extent that this trend might represent a growing international awareness of the threat represented by nuclear weapons including a hypothesized increasing reluctance of existing nuclear-weapon states to share their technology. The current process of ratifying the new {{w|Treaty on the Prohibition of Nuclear Weapons}} supports the hypothesis of such a trend, while the lack of universal support for it and the trend visible in Figure 1 clearly indicate that nuclear proliferation is still likely to continue.  We use this model to extend the 74 years of history of nuclear proliferation available as this is being written on 2020-04-21 into predicting another 74 years into the future.

== How did the existing nuclear-weapon states develop this capability? == 

There are, of course, multiple issues in nuclear proliferation:  a new nuclear-weapon state requires at least four distinct things to produce a nuclear weapon: motivation, money, knowledge, and material.  And many if not all of the existing nuclear-weapon states got foreign help, as outlined below and summarized in the accompanying table.  

'''Disclaimer''':  Complete answers to each of these questions for every nuclear-weapon state can never be known with certainty. The literature found by the present authors is summarized in the accompanying table with citations to the literature in the following discussion but should not be considered any more authoritative than the sources cited, some of which may not be adequate to support all the details and the generalizations in the accompanying table.

However, this analysis should be sufficient to support the general conclusions of this article.  

{| class=&quot;wikitable&quot;
|-
! rowspan=&quot;2&quot; | Country 
! rowspan=&quot;2&quot; | Motivation
! rowspan=&quot;2&quot; | Money 
! rowspan=&quot;2&quot; | Knowledge 
! rowspan=&quot;2&quot; | Material 
! colspan=&quot;2&quot; | Foreign Help 
|-
! Who 
! Why 
|-
| US 
| Nazi threat
| self
| own scientists + immigrants, esp. fr. Germany &amp; Italy in collaboration with the UK and Canada.  
| Congo + self 
| GB (incl. Canada)
| Nazi threat
|-
| USSR (RU)
| Hiroshima &amp; Nagasaki bombs + western invasions during WW II, after WW I, and before
| self
| own scientists + espionage in the US &amp; captured Germans
| self 
| US (espionage)
| US scientists wanted to protect USSR
|- 
| UK (GB)
| USSR
| self
| Manhattan Project
| Canada
| colspan=&quot;2&quot; style=&quot;text-align: center;&quot; | ?
|- 
| France (FR)
| USSR + Suez Crisis
| self
| self
| self
| colspan=&quot;2&quot; style=&quot;text-align: center;&quot; | ?
|- 
| China (CN) 
| 1st Taiwan Strait Crisis 1954–1955, the Korean Conflict, etc.
| self
| USSR
| self
| RU 
| US threat 
|- 
| India (IN)
| loss of territory in the China-Himalayan border dispute-1962
| self
| students in UK, US
| Canadian nuc reactor
| colspan=&quot;2&quot; style=&quot;text-align: center;&quot; | ?
|- 
| Israel (IL) 
| hostile neighbors
| self
| self + France
| France + ???
| colspan=&quot;2&quot; style=&quot;text-align: center;&quot; | ?
|- 
| rowspan=&quot;2&quot; | Pakistan (PK)
| rowspan=&quot;2&quot; | Loss of E. Pakistan in 1971
| rowspan=&quot;2&quot; | Saudis + self
| rowspan=&quot;2&quot; | US, maybe China?
| rowspan=&quot;2&quot; | self?
| US || USSR in Afghanistan
|-
| CN || ? 
|- 
| N.Korea (KP)  
| threats fr. US
| self?
| US via Pakistan?
| self?
| PK +? 
| ? 
|}

'''''Table 1.  Where did the existing nuclear-weapon states get the motivation, money, knowledge, and material for their nuclear-weapons program?'''''  

[[File:GDP of nuclear-weapon states (billions of 2019 USD).svg|thumb|Figure 2.  {{w|Gross Domestic Product}} (GDP) of nuclear-weapon states in billions of 2019 US dollars at {{w|Purchasing Power Parity}} (PPP) before (dashed line), during (thick solid line) and after (thinner solid line) their nuclear-weapons program leading to their first test of a nuclear weapon.  (Country codes as with Figure 1.)  The dotted line indicates the total cost of the Manhattan Project that developed the very first nuclear weapon from 1942 to the end of 1945.]]

To help us understand the differences in sizes of the different nuclear-weapon states, Figure 2 plots the evolution of GDP in the different nuclear-weapon states.  The following subsections provide analysis with references behind the summaries in Table 1 and Figure 2.

=== Motivation === 

Virtually any country that feels threatened would like to have some counterweight against aggression by a potential enemy.  

* The US funded the Manhattan project believing that Nazi Germany likely had a similar project.  

* Soviet leaders might have felt a need to defend themselves from nuclear coercion after having been invaded by Nazi Germany only a few years earlier, and having defeated [[w:Allied intervention in the Russian Civil War|foreign invasions from the West and the East after World War I trying to put the Tsar back in power]].&lt;ref&gt;{{cite Q|Q91370284}}&lt;!-- Fogelsong (1995) America's Secret War against Bolshevism: U.S. Intervention in the Russian Civil War, 1917-1920 --&gt;. That doesn't count [[w:|French invasion of Russia|numerous other invasions that are a sordid part of Russian history]], which educated Russians throughout history would likely remember, even if their invaders may not.&lt;/ref&gt;

* The United Kingdom and France felt nuclear threats from the Soviet Union.&lt;ref&gt;The UK and France would have had many reasons to fear the intentions of the USSR during the early period of the {{w|Cold War}}:  The first test of a nuclear weapon by the USSR came just over three months after the end of the 1948-49 {{w|Berlin Blockade}}.  Other aspects of Soviet repression in countries they occupied in Eastern Europe contributed to the failed {{w|Hungarian Revolution of 1956}}.&lt;/ref&gt; France's concern about the Soviets increased [[w:France and weapons of mass destruction#cite note-16|after the US refused to support them during the 1956]] {{w|Suez Crisis}}:  If the US would not support a British-French-Israeli invasion of Egypt, the US might not defend France against a possible Soviet invasion.&lt;ref&gt;{{cite Q|Q91382112}}&lt;!-- Devid Fromkin (2006) Stuck in the Canal, NYT--&gt;.  See also [[w:France and weapons of mass destruction]].&lt;/ref&gt; 

* China reportedly decided to initiate its nuclear weapons program during the [[w:China and weapons of mass destruction#Nuclear weapons|First Taiwan Strait Crisis of 1954-55]],&lt;ref&gt;[[w:China and weapons of mass destruction#Nuclear weapons]];  see also [[w:First Taiwan Strait Crisis]], {{cite Q|Q63874609}}&lt;!--  Morton Halperin (1966) The 1958 Taiwan Straits Crisis: A documentary history --&gt;, and [[w:Daniel Ellsberg]].&lt;/ref&gt; following nuclear threats from the US regarding Korea.&lt;ref&gt;{{cite Q|Q63874136}}&lt;!-- The Atomic Bomb and the First Korean War --&gt;.  See also [[w:Daniel Ellsberg]].&lt;/ref&gt;

* India lost territory to China in the 1962 {{w|Sino-Indian War}}, which reportedly convinced India to abandon a policy of avoiding nuclear weapons.&lt;ref&gt;{{cite Q|Q91391545}}&lt;!-- Bruce Riedel (2012) JFK's Overshadowed Crisis --&gt;.  See also {{w|India and weapons of mass destruction}}.  India and China have continued to have conflicts.  See, for example, the Wikipedia articles on [[w:China-India relations]] and the [[w:2017 China-India border standoff]].&lt;/ref&gt;

* Pakistan's nuclear weapons program began in 1972 in response to the loss of East Pakistan (now Bangledesh) in the 1971 {{w|Bangladesh Liberation War}}.&lt;ref&gt;{{w|Pakistan and weapons of mass destruction}}.  {{w|India-Pakistan relations}} have been marked by frequent conflict since the two nations were born with the dissolution of the British Raj in 1947.  This history might help people understand the need that Pakistani leaders may have felt and still feel for nuclear parity with India, beyond the loss of half their population and 15 percent of their land area in the 1971 Bangladesh Liberation War.&lt;/ref&gt; On November 29, 2016, Moeed Yusuf claimed that the threat of a nuclear war between India and Pakistan was the most serious foreign policy issue facing then-President-elect Trump.&lt;ref&gt;{{cite Q|Q91271615}}&lt;!-- Moeed Yusuf (2016-11-26) “An India-Pakistan Crisis:  Should we care?”, War on the Rocks --&gt;.&lt;/ref&gt; That may have been an overstatement, but the possibilities of a nuclear war between India and Pakistan should not be underestimated.  [[w:Indo-Pakistani wars and conflicts|There have been lethal conflicts between India and Pakistan at least as recent as 2019.]]  If that conflict goes nuclear, it could produce a “nuclear autumn” during which a quarter of humanity not directly impacted by the nuclear war would starve to death, according to simulations by leading climatologists.&lt;ref&gt;Helfand and the references he cited predicted two billion deaths.  With a [[w:world population|world population]] in 2013 of 7.2 billion, less than 8 billion, 2 billion is more than a quarter of humanity.  See &lt;!-- Nuclear famine: two billion people at risk? --&gt;{{cite Q|Q63256454}}.  See also Toon et al. (2007).&lt;/ref&gt;

* Israel has faced potentially hostile neighbors since its declaration of independence in 1948.&lt;ref&gt;{{w|Arab-Israeli conflict}}.  Threats perceived by Israel continue, including the {{w|2018 Gaza border protests}} that have continued at least into 2020.  One might therefore reasonably understand why Israel might feel a need for nuclear weapons and why others might believe that the 1979-09-22 {{w|Vela incident}} was an Israeli nuclear test.&lt;/ref&gt;

* North Korea first tested a nuclear weapon on 2006-10-09,&lt;ref&gt;{{cite Q|Q59596578}}&lt;!-- Jonathan Medalia (2016) Comprehensive Nuclear-Test-Ban Treaty: Background and Current Developments, Congressional Research Service --&gt;;  The US Congressional Research Service in 2016 reported, “The Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO) PrepCom's international monitoring system detected data indicating that North Korea had conducted a nuclear test on January 6, 2016. ... On October 9, 2006, North Korea declared that it had conducted an underground nuclear test.”  For the present purposes, we use the October date declared by North Korea, not the January date reported by CTBTO.  See also {{w|2006 North Korean nuclear test}}.&lt;/ref&gt; less than five years after having been named as part of an &quot;{{w|Axis of evil}}&quot; by US President George W. Bush on 2002-01-29.&lt;ref&gt;{{cite Q|Q91337578}}&lt;!-- 2002 State of the Union Address by US President George W. Bush--&gt;;  see also ''[[w:Axis of evil]]''.&lt;/ref&gt; Chomsky claimed that the relations between the US and North Korea have followed &quot;a kind of tit-for-tat policy.  You make a hostile gesture, and we'll respond with a crazy gesture of our own.  You make an accommodating gesture, and we'll reciprocate in some way.&quot;  He gave several examples including a 1994 agreement that halted North Korean nuclear-weapons development.  &quot;When George W. Bush came into office, North Korea had maybe one [untested] nuclear weapon and verifiably wasn't producing any more.&quot;&lt;ref&gt;{{cite Q|Q86247233}}&lt;!-- Who Rules the World?, 2017 book by Noam Chomsky--&gt;, pp. 131-134. Chomsky includes in this game of tit-for-tat the total destruction of North Korean infrastructure during the Korean War in the early 1950s, including huge dams that controlled the nation's water supply, destroying their crops, and raising the spectre of mass starvation.  {{cite Q|Q91455702}}&lt;!-- Report on the destruction of dikes: Holland 1944-45 and Korea 1953 --&gt; noted that German General Syss-Inquart ordered similar destruction of dikes in Holland in 1945, which condemned many Dutch civilians to death by starvation. For that crime Syss-Inquart became one of only 24 of the people convicted at the Nurenberg war crimes trial to have been sentencted to death.  Chomsky noted that this is &quot;not in our memory bank, but it's in theirs.&quot;&lt;/ref&gt;

All this suggests that it will be difficult to reduce the threat of nuclear proliferation and nuclear war without somehow changing the nature of international relations so weaker countries have less to fear from the demands of stronger countries.

=== Money ===

It's no accident that most of the world's nuclear-weapon states are large countries with substantial populations and economies. That's not true of Israel with only roughly 9 million people nor North Korea with roughly 26 million people in 2018.  France and the UK have only about 67 and 68 million people, but they are also among the world leaders in the size of their economies.  

Pakistan is a relatively poor country.  It reportedly received financial assistance from Saudi Arabia for its nuclear program.&lt;ref&gt;{{cite Q|Q84288832}}&lt;!-- Saudi Arabia: Nervously Watching Pakistan --&gt;.&lt;/ref&gt;   

Another reason for a possible decline in the rate of nuclear proliferation apparent in Figure 1 is the fact that among nuclear-weapon states, those with higher GDPs tended to acquire this capability earlier, as is evident in Figure 2.

=== Knowledge ===

In 1976, {{w|John Aristotle Phillips}}, an &quot;underachieving&quot; undergraduate at Princeton University, &quot;designed a nuclear weapon using publicly available books and papers.&quot;&lt;ref&gt;{{cite Q|Q91459264}}&lt;!-- Student Designs Nuclear Bomb (1976-10-09) Spokane Daily Chronicle--&gt;. See also [[w:John Aristotle Phillips]].&lt;/ref&gt; Nuclear weapons experts disagreed on whether the design would have worked.  Whether Phillips' design would have worked or not, it should be clear that the continuing progress in human understanding of {{w|nuclear physics}} inevitably makes it easier for people interested in making such weapons to acquire the knowledge of how to do so.  

Before that, the nuclear age arguably began with the 1896 discovery of radioactivity by the French scientist Henri Becquerel.  It was further developed by Pierre and Marie Curie in France, Ernest Rutherford in England, and others, especially in France, England and Germany.&lt;ref&gt;{{w|Nuclear physics}}.&lt;/ref&gt; In 1933 after Adolf Hitler came to power in Germany, {{w|Leo Szilard}} moved from Germany to England. The next year he patented the idea of a nuclear fission reactor.  Other leading nuclear scientists similarly left Germany and Italy for the UK and the US.  After World War II began, the famous {{w|Manhattan Project}} became a joint British-American project, which produced the very first test of a nuclear weapon.&lt;ref&gt;{{w|History of nuclear weapons}}.&lt;/ref&gt; 

After Soviet premier {{w|Joseph Stalin}} learned of the atomic bombings of Hiroshima and Nagasaki, the USSR (now Russia) increased the funding for their nuclear-weapons program.  That program was helped by intelligence gathering about the German nuclear weapon project and the American Manhattan Project.&lt;ref&gt;{{cite Q|Q91461780}}&lt;!-- Espionage and the Manhattan Project (1940-1945), Office of Scientific and Technical Information, US Department of Energy --&gt;.  See also {{w|Soviet atomic bomb project}}.&lt;/ref&gt;

The UK's nuclear-weapons program was built in part on their wartime participation in the Manhattan Project, as noted above.  

France was among the leaders in nuclear research until World War II.  They still had people with the expertise needed after the 1956 {{w|Suez Crisis}} convinced them they needed to build nuclear bombs, as noted above.&lt;ref&gt;See also {{w|History of nuclear weapons}}.&lt;/ref&gt;

China got some help from the Soviet Union during the initial phases of their nuclear program.&lt;ref&gt;{{w|China and weapons of mass destruction}}.&lt;/ref&gt;

The first country to get nuclear weapons after the Non-Proliferation Treaty was India.  Their Atomic Energy Commission was founded in 1948, chaired by {{w|Homi J. Bhabha}}.  He had published important research in nuclear physics while a graduate student in England in the 1930s, working with some of the leading nuclear physicists of that day.&lt;ref&gt;[[w: Homi J. Bhabha]];  see also [[w:Timeline of nuclear weapons development]].&lt;/ref&gt;

Meanwhile, Israel's nuclear weapons program initially included sending students abroad to study under leading physicists like Enrico Fermi at the University of Chicago.  It also included extensive collaboration with the French nuclear-weapons program.&lt;ref&gt;[[w:Nuclear weapons and Israel]].  See also [https://www.wisconsinproject.org/israels-nuclear-weapon-capability-an-overview/ &quot;Israel’s Nuclear Weapon Capability: An Overview&quot;], July 1, 1996, by the Wisconsin Project on Nuclear Arms Control.&lt;/ref&gt;

Pakistan got &quot;dual use&quot; production technology and complete nuclear-capable delivery systems from both the US and China.&lt;ref&gt;For Chinese help to Pakistan, see {{cite Q|Q95917195}}&lt;!--  Gradual Signs of Change: Proliferation to and from China over Four Decades --&gt;.&lt;/ref&gt; Pakistan got secret help from the US in the 1980s in violation of US law to secure Pakistani cooperation with US support for anti-Soviet resistance in Afghanistan.&lt;ref&gt;{{cite Q|Q91463994}}&lt;!-- New Documents Spotlight Reagan-era Tensions over Pakistani Nuclear Program, research report by William Burr, Wilson Center --&gt;. {{cite Q|Q91464530}}&lt;!-- Pakistan's Illegal Nuclear Procurement Exposed in 1987: Arrest of Arshed Pervez Sparked Reagan Administration Debate over Sanctions, National Security Archive Electronic Briefing Book No. 446 --&gt;. See also [[w:Pakistan and weapons of mass destruction]].&lt;/ref&gt; (In 1995 the Wisconsin Center on Nuclear Arms Control reported that Pakistan’s most reliable nuclear delivery platforms were French-made Mirage fighters,&lt;ref&gt;{{cite Q|Q95919096}}&lt;!-- Pakistan: American, Chinese or French Planes Would Deliver its Bomb, Wisconsin Project on Nuclear Arms Control --&gt;.&lt;/ref&gt; though they also had US-made F-16s they could modify to carry those weapons.)  

{{w|Abdul Qadeer Khan}}, a leader in Pakistan's nuclear weapons program, has also faced multiple allegations of being one of the world's leading nuclear proliferators in operating a black market in nuclear weapons technology.  North Korea, Iran and other countries have allegedly received help from Pakistan for their nuclear weapons programs with at least some of it coming via A. Q. Khan's black market dealings.&lt;ref&gt;A summary of this appears in [https://www.wisconsinproject.org/pakistan-nuclear-milestones-1955-2009/ &quot;Pakistan Nuclear Milestones, 1955-2009&quot;] by the {{w|Wisconsin Project on Nuclear Arms Control}}.  See also citations on this in the Wikipedia article on [[w:Abdul Qadeer Khan]].&lt;/ref&gt; Some of this technology was reportedly obtained from the US in the 1980s with the complicity of US government officials who wanted Pakistan's help for groups in Afghanistan fighting the Soviets.&lt;ref&gt;E.g., {{cite Q|Q88306915}}&lt;!-- Lyndsey Layton (7 July 2007), &quot;Whistle-Blower's Fight For Pension Drags On&quot;, The Washington Post --&gt;, and [[w:Richard Barlow (intelligence analyst)|Richard Barlow]].&lt;/ref&gt; 

{{w|Vikram Sood}}, a former head of India's foreign intelligence agency, said, &quot;America fails the IQ test&quot; in discussing A. Q. Khan's nuclear black market, adding that Pakistan ''may'' have given nuclear-weapons technology to al Qaeda &quot;just weeks prior to September 11, 2001.&quot;&lt;ref&gt;{{cite Q|Q88310866}}&lt;!-- America fails the IQ test--&gt;&lt;/ref&gt; It may not be wise to accept Sood's claim at face value, given the long-standing hostility between India and Pakistan. In April 2002 Milhollin, Founder and then Executive Director of the Wisconsin Project on Nuclear Arms Control, said that Al Qaeda &quot;is interested in getting weapons of mass destruction, [and if it] can organize a 19-person group to fly airliners into buildings, it can smuggle a nuclear weapon across a border.&quot;&lt;ref&gt;{{cite Q|Q95987528}}&lt;!-- Use of Export Controls to Stop Proliferation --&gt;.&lt;/ref&gt; In 2005 Robert Gallucci, a leading researcher and expert on nuclear proliferation who served in high level positions in the Reagan, G. H. W. Bush and Clinton administrations because of this expertise, wrote that there was an unacceptably high probability &quot;that Al Qaeda or one of its affiliates will detonate a nuclear weapon in a US city ... . The loss of life will be measured ... in the hundreds of thousands. ... Consider the more likely scenarios ... .  An Al Qaeda cell ... purchases 50 or so kilograms of highly enriched uranium.  Today, the sellers might be Pakistan or Russia;  tomorrow they might be North Korea or Iran.  ... Another scenario ... involves the acquisition ... of a completed nuclear weapon.&quot;&lt;ref&gt;Gallucci's estimate of the probability of a nuclear attack by a terrorist group has declined substantially since 2005.  Back then, he wrote that a terrorist attack with a nuclear weapon in the next five to ten years &quot;is more likely than not&quot;.  In a private communication on June 4, 2020, he wrote, &quot;I was wrong in my estimate [that such an attack was more likely than not], and glad that I was. I don't understand AQ to be the threat now that it was fifteen years ago, but my concern continues that it is principally the unavailability of fissile material that prevents a terrorist from constructing an improvised nuclear device.&quot; The quote from 2005 is available in {{cite Q|Q96062427}}&lt;!-- Averting Nuclear Catastrophe: Contemplating Extreme Responses to U.S. Vulnerability, Harvard International Review, 2005, pp. 84, 83 --&gt;.  Essentially this same quote appears in a longer article by the same name:  {{cite Q|Q29395474}}&lt;!--Averting Nuclear Catastrophe: Contemplating Extreme Responses to U.S. Vulnerability, Annals of the American Academy of Political and Social Science, 2006--&gt;.&lt;/ref&gt;
  
And the US is helping Saudi Arabia obtain nuclear power, in spite of (a) the evidence that [[w:The 28 pages|the Saudi government including members of the Saudi royal family were involved at least as early as 1999 in preparations for the suicide mass murders of September 11, 2001]],&lt;ref&gt;{{cite Q|Q1702537}}&lt;!-- Joint inquiry into intelligence community activities before and after the terrorist attacks of September 11, 2001 --&gt;.  See also {{w|The 28 pages}}, which were redacted from the official report published 2003-01-29 and declassified in July 2016 by then-President Obama.&lt;/ref&gt; and (b) their [[w:Saudi Arabian-led intervention in Yemen|on-going support for Al Qaeda in Yemen, reported as recently as 2018]].&lt;ref name=SaudiQaeda&gt;{{cite Q|Q61890713}}&lt;!-- AP Investigation: US allies, al-Qaida battle rebels in Yemen--&gt;.&lt;/ref&gt;

=== Material === 

Reportedly the most difficult part of making nuclear weapons today is obtaining sufficient fissile material.  Toon et al. (2007) said, &quot;Thirteen countries operate plutonium and/or uranium enrichment facilities, including Iran&quot;, but Iran did not have sufficient fissile material in 2003 to make a nuclear weapon.  Another 20 were estimated to have had sufficient stockpiles of fissile material acquired elsewhere to make nuclear weapons.  They concluded that 32 (being 13 minus 1 plus 20) additional countries have sufficient fissile material to make nuclear weapons if they want.&lt;ref&gt;pp. 1975, 1977.  The 32 countries they identified included 12 of the 13 that &quot;operate plutonium and/or uranium enrichment facilities&quot;, excepting Iran as noted.  The other 20 countries acquired stockpiles elsewhere. In addition to the 32 with sufficient fissile material to make a nuclear weapon, Egypt, Iraq and the former Yugoslavia were listed as having abandoned a nuclear-weapons program.&lt;/ref&gt;

Toon et al. (2007) also said, &quot;In 1992 the International Atomic Energy Agency safeguarded less than 1% of the world’s HEU [Highly Enriched Uranium] and only about 35% of the world inventory of Pu [Plutonium] ... .  Today [in 2007] a similarly small fraction is safeguarded.&quot; 

HEU is obtained by separating &lt;sup&gt;235&lt;/sup&gt;U, which is only 0.72 percent of naturally occurring uranium.&lt;ref&gt;{{cite Q|Q91488549}}&lt;!-- Weapons of Mass Destruction (WMD):  Uranium Isotopes --&gt;.&lt;/ref&gt; Weapons-grade uranium has at least 85 percent &lt;sup&gt;235&lt;/sup&gt;U.&lt;ref&gt;See the section on “Highly enriched uranium (HEU)” in the Wikipedia article on [[w:Enriched uranium]].&lt;/ref&gt; Thus, at least 0.85/0.0072 = 118 kg of naturally occurring uranium are required to obtain 1 kg that is weapons-grade.  Toon et al. (2007) estimated that 25 kg of HEU would be used on average for each &lt;sup&gt;235&lt;/sup&gt;U-based nuclear weapon.  Plutonium, by contrast, is a byproduct of energy production in standard &lt;sup&gt;238&lt;/sup&gt;U nuclear reactors. 

Much of the uranium for the very first test of a nuclear weapon by the US came from the Congo,&lt;ref name='Ures'&gt;[[w:Manhattan project]].&lt;/ref&gt; but domestic sources provided most of the uranium for later US nuclear-weapons production.&lt;ref&gt;[[w:List of countries by uranium reserves]].&lt;/ref&gt; The Soviet Union (USSR, now Russia) also seems to have had adequate domestic sources for its nuclear-weapons program, especially including Kazakhstan, which was part of the USSR until 1990;  Kazakhstan has historically been the third largest source of uranium worldwide after Canada and the US.&lt;ref name='Ures'/&gt; The UK presumably got most of its uranium from Canada.  

The French nuclear-weapons program seems to have been built primarily on plutonium.&lt;ref&gt;{{w|France and weapons of mass destruction}}.  See also Table 2 in Toon et al. (2007), which claims that in 2003, France had enough fissile material for roughly 24,000 plutonium bombs and 1,350 &lt;sup&gt;235&lt;/sup&gt;U bombs.&lt;/ref&gt; This required them to first build standard &lt;sup&gt;238&lt;/sup&gt;U nuclear reactors to make the plutonium.  Then they didn't need nearly as much uranium to sustain their program.  

China has reportedly had sufficient domestic reserves of uranium to support its own needs,&lt;ref name='Ures'/&gt; even exporting some to the USSR in the 1950s in exchange for other assistance with their nuclear defense program.&lt;ref&gt;[[w:China and weapons of mass destruction]].&lt;/ref&gt; 

India's nuclear weapons program seems to have been entirely (or almost entirely) based on plutonium.&lt;ref&gt;[[w:India and weapons of mass destruction]];  see also Toon et al. (2007) and [[w:List of countries by uranium reserves]].&lt;/ref&gt; 

Israel seems not to have had sufficient uranium deposits to meet its own needs.  Instead, they purchased some from France until France ended their nuclear-weapons collaboration with Israel in the 1960s. To minimize the amount of uranium needed, nearly all Israeli nuclear weapons seem to be plutonium bombs.&lt;ref&gt;Toon et al. (2007).&lt;/ref&gt;

It's not clear where Pakistan got most of its uranium: Its reserves in 2015 were estimated at zero, and its historical production to that point was relatively low.&lt;ref name='Ures'/&gt; By comparison with the first seven nuclear-weapon states, it's not clear where Pakistan might have gotten enough uranium to produce 83 plutonium bombs and 44 uranium bombs, as estimated by Toon et al. (2007, Table 2, p. 1976.)  As previously noted, the US helped the Pakistani nuclear-weapons program in the 1980s and accused China of providing similar assistance, a charge that China has repeatedly and vigorously denied.  China has provided civilian nuclear reactors, which could help produce plutonium but not &lt;sup&gt;235&lt;/sup&gt;U.&lt;ref&gt;[[w:Pakistan and weapons of mass destruction#Alleged foreign co-operation]].&lt;/ref&gt;

According to the Federation of American Scientists, &quot;North Korea maintains uranium mines with an estimated four million tons of exploitable high-quality uranium ore ... that ... contains approximately 0.8% extractable uranium.&quot;&lt;ref&gt;{{cite Q|Q91520731}}&lt;!--DPRK: Nuclear Weapons Program per the Federation of American Scientists--&gt;.  See also [[w:North Korea and weapons of mass destruction]].&lt;/ref&gt; If that's accurate, processing all that would produce 4,000,000 times 0.008 = 32,000 tons of pure natural uranium, which should be enough to produce the weapons they have today.  

=== Conclusions regarding motivation, money, knowledge, and material ===

1. There seems to be no shortage of motivations for other countries to acquire nuclear weapons.  The leaders of the Soviet Union had personal memories of being invaded not only by Germany during World War II but also by the US and others after World War I.  The UK had reason to fear the Soviets in their occupation of Eastern Europe.  The French decided after Suez they couldn't trust the US to defend them.  China had been forced to yield to nuclear threats before starting their nuclear program, as did India, Pakistan and North Korea.  Israel has fought multiple wars since their independence in 1948.  
  
2. The knowledge and material required to make such weapons in a relatively short order are also fairly widely available, even without the documented willingness of current nuclear powers to secretly help other countries acquire such weapons in some cases.&lt;ref&gt;In addition to the 32 currently non-nuclear-weapon states with &quot;sufficient fissile material to make nuclear weapons if they wished&quot;, per Toon et al. (2007), the inspector general of the US Department of Energy concluded in 2009 (in its most recent public accounting) that enough highly enriched uranium was missing from US inventories to make at least five nuclear bombs comparable to those that destroyed substantial portions of Hiroshima and Nagasaki in 1945.  The issue of missing fissile material is likely much larger than what was reported missing from US inventories, because substantially more weapons-grade material may be missing in other countries, especially Russia, as noted by {{cite Q|Q91521732}}&lt;!-- Plutonium is missing, but the government says nothing --&gt;.&lt;/ref&gt;
  
3. Unless there is some fundamental change in the structure of international relations, it seems unwise to assume that there will not be more nuclear-weapon states in the future, with the time to the next &quot;first test&quot; of a nuclear weapon following a probability distribution consistent with the previous times between &quot;first tests&quot; of nuclear weapons by the current nuclear-weapon states.

== Distribution of the time between Poisson “first tests” ==

Possibly the simplest model for something like the time between &quot;first tests&quot; in an application like this is to assume they come from one {{w|exponential distribution}} with 8 observed times between the 9 current nuclear-weapon states plus one [[w:Censoring (statistics)|censored observation]] of the time between the most recent one and a presumed next one. This simple theory tells us that the maximum likelihood estimate of the mean time between such &quot;first tests&quot; is the total time from the US &quot;Trinity&quot; test to the present, 74.8 years, divided by the number of new nuclear-weapon states, 8, not counting the first, which had no predecessors.  Conclusion:  Mean time between &quot;first tests&quot; = 9.3 years.&lt;ref&gt;For precursors to the current study that involve censored estimation of time to a nuclear war, see [[Time to extinction of civilization]] and [[Time to nuclear Armageddon]].&lt;/ref&gt;

However, Figure 1 suggests that the time between &quot;first tests&quot; of succeeding nuclear-weapon states is increasing.  The decreasing hazard suggested by this figure requires mathematics that are not as easy as the censored data estimation as just described. 

[[File:NucWeaponStates logYrsBetw1stTsts.svg|thumb|Figure 3.  Semilog plot of the years between new nuclear-weapon states.  (Country codes as with Figure 1.)]]

To understand the current data better, we redo Figure 1 with a log scale on the y axis in Figure 3.   

Figures 1 and 3 seem consistent with the following:

* If the mean time between &quot;first tests&quot; is increasing over time, as suggested by Figures 1 and 3, then the distribution cannot be exponential, because that requires a constant [[w:Survival analysis#Hazard function and cumulative hazard function|hazard rate]].&lt;ref&gt;For the exponential distribution, &lt;math&gt;h(t) = (-d/dt \log S(t)) = \lambda&lt;/math&gt;, writing the exponential survival function as &lt;math&gt;S(t) = \exp(-\lambda t)&lt;/math&gt;.&lt;/ref&gt;

* Even though nuclear proliferation has been slowing since 1950, it seems not to have slowed fast enough to support the assumption that nuclear proliferation can be ignored, which seems to be implied by current international policy.  It could ''accelerate'' in the future if more states began to perceive greater threats from other nations.  

* Fortunately we can simplify this modeling problem by using the famous duality between exponential time between events and a Poisson distribution for numbers of events in specific intervals of time.  By modeling Poisson counts of &quot;first tests&quot; each year, we can use techniques for Poisson regression for models suggested by Figure 3.  The simplest such model might consider log(Poisson mean numbers of &quot;first tests&quot; each year) to be linear in the time since the first test of a nuclear weapon (code-named [[w:Trinity (nuclear test)|&quot;Trinity&quot;]]).&lt;ref&gt;{{cite Q|Q7749726}}&lt;!-- Richard Rhodes (1986) The Making of the Atomic Bomb --&gt;.  See also [[w:Trinity (nuclear test)]].&lt;/ref&gt;  

* The image in Figure 3 suggests the time between “first tests” by new nuclear-weapon states may be increasing, but not necessarily liearly.  Easily tested alternatives to linearity could be second, third and fourth powers of the &quot;timeSinceTrinity&quot;.&lt;ref&gt;One might also consider a model with the log(Poisson mean) behaving like a [[w:Wiener process|&quot;Wiener process&quot; (also called a &quot;Brownian motion&quot;)]].  This stochastic formulation would mean that the variance of the increments in log(hazard) between &quot;first tests&quot; is proportional to the elapsed time.  See {{cite Q|Q91547149}}&lt;!-- Wolfram: Wiener Process--&gt; and [[w:Wiener process]].  The “bssm” package for R should provide a reasonable framework for modeling this;  see {{cite Q|Q91626942}}&lt;!-- bssm: Bayesian Inference of Non-Linear and Non-Gaussian State Space --&gt;.  However, this author's efforts to use this package for this purpose have so far produced unsatisfactory results.  More time understanding the software might produce better results but not necessarily enough better to justify the effort that might be required.&lt;/ref&gt;

We used Poisson regression to model this as a series of the number of events each year.&lt;ref&gt;We could have used one observation each month, week, or day.  Such a change might give us slightly better answers while possibly increasing the compute time more than it's worth.&lt;/ref&gt;

== Parameter estimation == 

For modeling and parameter estimation, we model the number of “first tests” of a new nuclear-weapon state each year (1 in 8 years, 0 in the remaining 66 years between 1945 and 2019) with log(Poisson mean number of “first tests” each year) as polynomials in “timeSinceTrinity” = the time in years since the [[w:Trinity (nuclear test)|Trinity test by the US]], 1945-07-16.  The standard {{w|p-value}} for the {{w|Wald test}} of the linear model was 0.21 -- ''not'' statistically significant.  

{{w|George Box}} famously said that, [[w:All models are wrong|''&quot;All models are wrong, but some are useful.&quot;'']].&lt;ref&gt;{{cite Q|Q91658340}}&lt;!-- Empirical Model-Building and Response Surfaces --&gt;.&lt;/ref&gt;

Burnham and Anderson (1998) and others claim that better predictions can generally be obtained using Bayesian Model Averaging (BMA).&lt;ref&gt;See also {{cite Q|Q91670340}}&lt;!-- Bayesian model selection in social research, Adrian Raftery 1995 --&gt; and {{cite Q|Q62568358}}&lt;!-- Model selection and model averaging, Claeskens and Hjort, 2008 --&gt;.&lt;/ref&gt; In this case, we have two models:  log(Poisson mean) being constant or linear in “timeSinceTrinity”.  It is standard in the BMA literature to assume a priori an approximate uniform distribution over all models considered with a penalty for estimating each additional parameter to correct for the tendency of the models to overfit the data.  With these standard assumptions, this comparison of these two models estimated a 21 percent posterior posterior probability for the model linear in  “timeSinceTrinity”, leaving 79 percent probability for the model with a constant Poisson mean.  

[[File:NucWeaponStates BMAyrsBetw1stTsts.svg|thumb|Figure 4.  BMA constant-linear and quartic fits to time between new nuclear-weapon states.  (Country codes as with Figure 1.)]]

We also experimented with fitting up to quartic models in “timeSinceTrinity”.&lt;ref&gt;{{cite Q|Q91674106}}&lt;!-- BMA: Bayesian Model Averaging package for R --&gt;.  The algorithm used for this retained only the intercept and the coefficient of the highest power in each order.  Models like &lt;math&gt;b_0 + b_1 x + b_2 x^2&lt;/math&gt; with &lt;math&gt;b_1 \ne 0&lt;/math&gt; were considered but had a posterior probability so low they were not retained in the final mixture of models.  The quartic mixture retained only &lt;math&gt;b_0&lt;/math&gt; (constant), &lt;math&gt;b_0 + b_1 x&lt;/math&gt; (linear), &lt;math&gt;b_0 + b_2 x^2&lt;/math&gt; (quadratic), &lt;math&gt;b_0 + b_3 x^3&lt;/math&gt; (cubic), and &lt;math&gt;b_0 + b_4 x^4&lt;/math&gt; (quartic) with posterior probabilities 49.59, 13.24, 13.21, 12.66, and 12.31 percents, respectively.&lt;/ref&gt; These prediction lines were added to Figure 3 to produce Figure 4.  

Comparing predictions between the constant-linear and constant-quartic mixtures might help us understand better the limits of what we can learn from the available data.  A visual analysis of the right (quartic mixture) panel in Figure 4 makes one wonder if the quartic, cubic and quadratic fits are really almost as good as the linear, as suggested by minor differences in the posterior probabilities estimated by the algorithm used.  

However, the forecasts of nuclear proliferation will be dominated by the constant component of the BMA mixture.  Its posterior probability is 79 percent for the constant-linear mixture and 48.59 percent for the quartic mixture.  That means that the median line and all the lower quantiles of all simulated futures based on these models would be dominated by that constant term.  

Moreover, the quartic, cubic and quintic lines in the right (quartic mixture) panel of Figure 4 do not look nearly as plausible, at least to the present author, as the constant and linear lines.&lt;ref&gt;Recall that the estimation methodology here is Poisson regression, not ordinary least squares.&lt;/ref&gt; That, in turn, suggests that the constant linear mixture may be more plausible than the quartic mixture.  

We then used [[w:Monte Carlo method|Monte Carlo simulations]] with 5,000 random samples to compute central 60 and 80 percent confidence limits for the mean plus 80 percent prediction, and (0.8, 0.8) tolerance limits for future nuclear proliferation, as discussed in the next three sections of this article.&lt;ref name='cipiti'&gt;”{{w|Confidence intervals}}&quot; bound the predicted mean number of nuclear-weapon states for each future year considered.  Central 80 percent “{{w|prediction intervals}}&quot; are limits that include the central 80 percent of distribution of the number of nuclear-weapon states.  They add the uncertainty in the modeled Poisson process to the uncertainty of estimating the mean of that process for each future year considered.  We will also compute (0.8, 0.8) &quot;[[w:tolerance interval#Relation to other intervals|tolerance  intervals]]&quot;; &lt;math&gt;(p, 1-\alpha)&lt;/math&gt; tolerance intervals have a probability of &lt;math&gt;(1-\alpha)&lt;/math&gt; of containing a proportion of at least &lt;math&gt;p&lt;/math&gt; of all future observations.&lt;/ref&gt;

== Confidence limits == 

[[File:NucWeaponStates FcstMeanTimeBetw1stTsts.svg|thumb|Figure 5.  Confidence limits for the mean time between “first tests,” past and future per the constant-linear mixture.  (Country codes as with Figure 1.)]]

To get confidence limits, we simulated 5,000 Poisson mean numbers of &quot;first tests&quot; by new nuclear-weapon states for each of the 74 years used in the two BMA fits and another 74 years beyond.  These simulations were later used to compute confidence limits for the model estimates of the Poisson mean and prediction and tolerance limits for the actual number of nuclear-weapon states.&lt;ref name='cipiti'/&gt; First, however, we inverted the simulated Poisson means to get simulated exponential times, then summarized them to get simulated mean, median, and 60 and 80 percent confidence limits of the mean time to the next new nuclear weapon state.  We then added those simulation summary statistics from the constant-linear model in Figure 3 to produce Figure 5.  

The fairly flat shape of the median and lower 10 and 20 percent lines in Figure 5 seem consistent with a model that is a sum of a mixture of log-normal distributions with the dominant component having a posterior probability of either 79 or 48.59 percent and a constant mean, as noted in Figure 4.  The substantial curvature of the solid line forecast looks hopeful, with a mean of simulated means for the constant-linear mixture being almost 200 years between successive &quot;first tests&quot; by new nuclear-weapon states by the end of the forecasted period, 2093.  

The fact that the mean of the simulations exceeds the upper confidence limit for 2093 seems odd but can be explained by noting that this is a mixture of log-normal distributions, and the mean of a log-normal can exceed any quantile of its distribution if the standard deviation is sufficiently large.&lt;ref&gt;This follows, because quantile &lt;math&gt;q&lt;/math&gt; of a log-normal is &lt;math&gt;\exp[\mu+\sigma\Phi^{-1}(q)]&lt;/math&gt; and the mean is &lt;math&gt;\exp[\mu+\sigma^2/2]&lt;/math&gt;, so the mean exceeds quantile &lt;math&gt;q&lt;/math&gt; whenever &lt;math&gt;\sigma\Phi^{-1}(q) &lt; \sigma^2/2&lt;/math&gt;, i.e., when &lt;math&gt;\Phi^{-1}(q) &lt; \sigma/2&lt;/math&gt;.&lt;/ref&gt;

[[File:NucWeaponStates QuarticFcstMeanTimeBetw1stTsts.svg|thumb|Figure 6:  Confidence limits for the mean time between “first tests,” past and future per the quartic mixture.  (Country codes as with Figure 1.)]]

Note further that the distribution for each year in Figure 5 is a mixture of log-normal distributions, which means that their reciprocals, the mean numbers of &quot;first tests&quot; each year, will also be a mixture of log-normals with the same standard deviations on the log scale.  This standard deviation is larger the farther we extrapolate into the future.  

The increase over time in the ''mean'' time between &quot;first tests&quot; in Figures 5 and 6 suggests a desirable decrease in the rate of nuclear proliferation.  

However, we are more concerned with the ''shorter'' times between &quot;first tests&quot;, and they seem all too probable, as we shall see when we simulate and compute their cumulative sums.  To do that, we append these simulated predictions to a plot of the evolution of the number of nuclear-weapon states through the historical period.&lt;ref&gt;In these simulations, we assume a zero probability of a nuclear power giving up their nuclear weapons, even though [[w:South Africa and weapons of mass destruction|South Africa reportedly discontinued their nuclear weapons program in 1989]], prior to its [[w:South_Africa#End of apartheid|first universal elections in 1994]].  We could potentially add South Africa to our dataset of nuclear weapon states with the same date as Israel, then model the distribution of the time to when a nuclear-weapon state gives up its nuclear weapons using an exponential distribution.  For that, we have one observed time and eight such times that are censored.  Standard theory in that case says that the maximum likelihood estimate of the mean time to relinquishing nuclear weapons assuming an exponential distribution is the sum of all the times, censored or observed, divided by the number of times observed, not including the censored times in the denominator. For purposes of illustration, we will assume that South Africa dismantled its nuclear weapons 1989-12-31, though a report of an inspection by the International Atomic Energy Agency dated 1994-08-19 said they had dismantled six nuclear weapons and were still working to dismantle one more.  Based on this, the mean lifetime of a nuclear-weapon state can be estimated at 493 years.  We could potentially add this to the current modeling effort, but it would not likely change the answers enough to justify the additional effort.&lt;/ref&gt;

[[File:NucWeaponStates nucProlifPred.svg|thumb|Figure 7: Confidence limits on the number of nuclear-weapon states, past and predicted mean;  BMA constant-linear model on the left; quartic fit on the right. ]]

These numbers are plotted in Figure 7 for both BMA models considered.  The slope of the median lines are steeper than the recent history, but the statistical evidence does not support the naive interpretation of a slowing in nuclear proliferation that one might get from considering only the most recent data.  

Comparing the forecasts between the constant-linear and quartic BMA mixtures shows that the higher order quartic mixture widens the confidence limits, making the 20th percentile essentially flat with almost no additional nuclear proliferation, while the mean quickly escapes the upper limit.  That sharply rising mean suggests that less than 10 percent of the simulations predict nuclear arms races that involve many nation states and many more non-state armed groups.  These outcomes are not likely, but the probabilities of such outcomes seem too large to be dismissed without further consideration, especially when gambling with the future of civilization.  (Replications of these simulations with different sets of random numbers confirmed the stability of the images in Figure 7.)  

Ignoring the simulations of uncontrolled nuclear arms races, the median lines in Figure 7 predict between 16.3 and 14.5 at the end of the current simulated period, 2093, adding either 7.3 and 5.5 (for the constant-linear and quartic mixtures, respectively) to the current 9 nuclear-weapon states.  Those ''median'' numbers are a little less than double the number of nuclear-weapon states today.  

We extend this analysis by adding prediction intervals to these plots.

== Prediction limits == 

[[File:NucWeaponStates nucProlifPredInt.svg|thumb|Figure 8:  Number of nuclear-weapon states, past and predicted;  BMA constant-linear model on the left; quartic fit on the right.]]

The simplest bounds on the future are {{w|prediction interval}}s, which combine the statistical uncertainty in the estimates of mean numbers of nuclear-weapon states with the random variability in the outcomes.  We simulated 80 percent equal-tailed prediction limits and added them to Figure 7 to produce Figure 8.  

For both Bayesian mixture models, the most likely scenarios, especially the median line and the space between the 60 percent confidence limits, predict a continuation of nuclear proliferation.  It's difficult to imagine how that could continue without also substantively increasing the risk of nuclear war and therefore also of the extinction of civilization.  

[[File:NucWeaponStates nucProbs.svg|thumb|Figure 9:  Probabilities of the time to the next 5 new nuclear-weapon states using the constant-linear and quartic BMA models (left and right panels, respectively).]]

We can also summarize the simulations to estimate the probabilities of having 1, 2, 3, 4, and 5 new nuclear weapon states for each year in the prediction period between 2020 and 2093 in Figure 9.  This is another way of evaluating the sensibility of pretending there will be no further nuclear proliferation:  Not likely.  

Ninety-four percent of the simulations per the constant-linear model had at least one more nuclear-weapon state by 2093 and a 40 percent chance of at least 1 by 2025.  The quartic model predicts a 73 percent chance of at least one more nuclear-weapon state by 2093 and a 29 percent chance of at least one by 2024.  

The conclusions from both models include the following:  

:'''''The current structure of international relations'''''
:'''''seems to threaten the extinction of civilization.'''''

To better quantify the uncertainty in modeling, we can also construct tolerance intervals for the time to the next new nuclear-weapon state.

== Tolerance limits == 

[[File:NucWeaponStates nucProlifTolInt.svg|thumb|Figure 10:  Number of nuclear-weapon states with prediction and tolerance limits;  BMA constant-linear model on the left; quartic fit on the right.]]

We want to add statistical tolerance limits to Figure 8 in addition to the prediction limits.  To do this, we add Poisson simulations to the 80 percent confidence limits in Figure 7 rather than adding Poisson simulations to ''all the individual'' simulations summarized in Figure 7 to produce Figure 8.  The results appear in Figure 10.  

The upper limit lines in Figure 10 are higher than those in Figure 8.  It gives us a bit more humility regarding the value of current knowledge.  However, the difference is not enough to substantively alter our conclusions, namely that nuclear proliferation is likely until something makes it impossible for anyone to make more nuclear weapons for a very long time.  

== Discussion == 

A growing number of leading figures have said that as long as the world maintains large nuclear arsenals, it is only a matter of time before there is a nuclear war.  Concerns like this have been expressed by two former US Secretaries of Defense ({{w|Robert McNamara}}&lt;ref&gt;{{cite Q|Q64736611}}&lt;!-- Robert McNamara and James G. Blight (2003) Wilson's ghost:  Reducing the risk of conflict, killing, and catastrophe in the 21st century --&gt;.&lt;/ref&gt; and {{w|William Perry}}, two former US Secretaries of State {{w|Henry Kissinger}} and {{w|George Schultz}}, former US Senator {{w|Sam Nunn}}&lt;ref&gt;{{cite Q|Q92101045}}&lt;!-- George P. Shultz, William J. Perry, and Sam Nunn, “The Threat of Nuclear War Is Still With Us”, WSJ 2019-04-10--&gt;.&lt;/ref&gt; and others with, for example, the {{w|Nuclear Threat Initiative}}. {{w|Daniel Ellsberg}} has said that a nuclear war will most likely generate a nuclear winter that lasts several years during which 98 percent of humanity will starve to death if they do not die of something else sooner.&lt;ref&gt;{{cite Q|Q64226035}}&lt;!-- Daniel Ellsberg and Amy Goodman and Juan González, “Daniel Ellsberg Reveals He was a Nuclear War Planner, Warns of Nuclear Winter &amp; Global Starvation”, 2017-12-06, Democracy Now!--&gt;.&lt;/ref&gt;

Banerjee and Duflo, two of the three who won the 2019 Nobel Memorial Prize in Economics, have noted that neither economic nor political stability are assured for any country, including the United States, China and India.  In particular, they predict that economic growth will almost certainly slow substantially in the latter two, leaving many poor people in desperate economic straits.&lt;ref&gt;{{cite Q|Q85764011}}&lt;!-- Abhijit V. Banerjee and Esther Duflo, Good Economics for Hard Times, 2019--&gt;. Various journalists and academic researchers have expressed concern about increases in ethic violence in various countries and whether electoral transitions of power will continue, even in the US.  See, e.g., {{cite Q|Q92101761}}&lt;!-- Brian Klaas, “Everyone knows the 2020 election will be divisive. But will it also be violent?”, Washington Post, 2019-09-05--&gt;.&lt;/ref&gt; Internal problems in the US, China, India or any other nuclear-weapon state could push political leaders to pursue increasingly risky foreign adventures, like Argentina did in 1982,&lt;ref&gt;{{w|Falklands War}}.&lt;/ref&gt; possibly leading to a war that could produce [[Time to nuclear Armageddon|nuclear Armageddon]].&lt;ref&gt;The risks of a nuclear war producing major global climate problems have been documented in a series of simulations published in refereed academic journals, each more detailed and more disconcerting than the previous.  All assume that many firestorms will be produced, because (a) the areas targeted will likely be much more susceptible to firestorms than the underground or isolated sites used to test nuclear weapons, and (b) many of the weapons used will have yields substantially greater than those employed in Hiroshima and Nagasaki.  For a discussion of that literature, see [[Time to extinction of civilization]] and [[Time to nuclear Armageddon]].&lt;/ref&gt;

The evidence compiled in the present work only seems to increase the urgency of limiting the threat of nuclear war and nuclear proliferation in particular.

In the 20 years following the first test of a nuclear weapon on 1945-07-16 by the US, four more nations acquired such weapons.  In the 50 years since the Non-Proliferation Treaty took effect in 1970, another four acquired them.&lt;ref&gt;This uses a commonly accepted list of existing nuclear-weapon states and when they each first tested a nuclear weapon.  The sources used for the data are in the help file for the “nuclearWeaponStates” dataset in the “Ecdat” package for R.  See {{cite Q|Q56452356}}&lt;!-- Ecdat: Data Sets for Econometrics --&gt;.&lt;/ref&gt; Our analysis of the available data considering only the dates of these first tests suggests that nuclear proliferation may have been slowing throughout this period.  However, that apparent trend was not statistically significant in the model we fit.  

Bayesian Model Averages (BMA) is known to generally produce better predictions than single model fits.  Accordingly, we've estimated confidence, prediction, and tolerance limits for the number of new nuclear-weapon states 74 years into the future based on two BMA models with mixtures of either a constant with a linear model or a constant with terms up to quartic in the time since the very first test of a nuclear weapon.   

We can expect that some non-nuclear nations and terrorist groups would eagerly pursue nuclear weapons if such seemed feasible unless some unprecedented change in international law provided them with effective nonviolent recourse to perceived threats.{{cn|cite Bacevich: Mutual Assured Destruction may not deter someone who thinks that Armageddon might be good.}}  

Moreover, these weapons will likely become more available with the passage of time unless (a) a nuclear war destroys everyone's ability to make more such weapons for a long time, or (b) there is a major change in the structure of international relations that has far more success than similar previous efforts in limiting the ability of new nations and non-state actors to acquire nuclear weapons.

== Monitoring nuclear proliferation ==

Organizations like the {{w|Wisconsin Project on Nuclear Arms Control}}, the {{w|Federation of American Scientists}}, the {{w|Stockholm International Peace Research Institute}}, and other similar organizations seem to have made substantive contributions to the apparent reduction in the rate of nuclear proliferation visible in most of the plots included in this article.  

In 2017 the Nuclear Verification Capabilities Independent Task Force of the {{w|Federation of American Scientists}} published seven recommendations for improving the process of nuclear monitoring and verification:&lt;ref&gt;{{cite Q
|Q97136193
}}&lt;/ref&gt;

# A network of 4-5 independent Centers of Nonproliferation Authentication
# The {{w|P5+1}} and Iran should publicize important implementation steps.
# Periodic public updates on monitoring &amp; U.S. support to the IAEA.
# The P5+1 and others should encourage Iranian openness.
# A trusted body of outside experts should be created to monitor the Iranian nuclear agreement.
# NGOs concerned with nonproliferation should aggressively protect both the information and the physical safety of its sources.
# Funders of nonproliferation NGOs should strengthen cyber security.

These seven recommendations seem likely to contribute to the trend towards a reduction in the rate of nuclear proliferation visible in many of the figures included in this article.  

However, the Federation of American Scientists was founded in 1946, and only one of the current nuclear-weapon states had such weapons before they were founded.  When Stockholm International Peace Research Institute was founded in 1966, there were five nuclear-weapon states.  When the Wisconsin Project on Nuclear Arms Control was founded in 1986, there were seven.  Two more nations have joined the list of nuclear-weapon states since this Wisconsin Project was founded.  Even if these seven recommendations are fully implemented, it seems unlikely that those actions by themselves will end nuclear proliferation.  We can hope that they will contribute slowing the rate of nuclear proliferation already implicitly considered in the model fit and forecasts discussed above.  Sadly the recent actions by the US and Russia in embarking on major &quot;modernization&quot; programs seem to be cause for concern.

== The Trump administration and nuclear weapons == 

Several actions of the [[w:Presidency of Donald Trump|Trump administration]] have raised concern about a new arms race, escalating bellecosity of the US in international relations and even possibly accelerating the threat of further nuclear proliferation.

* Terminated the 1987-88 {{w|Intermediate-Range Nuclear Forces Treaty}} alleging both Russian non-compliance and concerns about the continuing growth of China's missile forces.  This was announced 20 October 2018 and completed on 1 February 2019.  

* On May 21, 2020, President Trump announced that the United States would withdraw from the 2002 {{w|Treaty on Open Skies}}, alleging Russian violations.&lt;ref name='openSkies'&gt;{{Cite news
|last=Riechman
|first=Deb
|date=May 21, 2020
|title=US says it's pulling out of Open Skies surveillance treaty
|publisher=[[Associated Press]]
|url=https://apnews.com/773c5e6b7fb92f5e6d0e4b8fddf1665e
|url-status=live
|access-date=May 21, 2020
|archive-url=https://web.archive.org/web/20200521151337/https://apnews.com/773c5e6b7fb92f5e6d0e4b8fddf1665e
|archive-date=May 21, 2020
}}&lt;/ref&gt;

* The 1996 {{w|Comprehensive Nuclear-Test-Ban Treaty}} has been signed but not ratified by the US.  This makes it legal for the [[w:Presidency of Donald Trump|Trump administration]] to resume testing at any time.  They have reportedly been discussing conducting the first nuclear test since 1992.&lt;ref name='openSkies'/&gt;

* The Trump administration has expressed a desire to build smaller more &quot;usable&quot; nukes.  The use of such weapons by themselves seem less likely to produce a {{w|nuclear winter}} or autumn but could increase the chances of a full scale nuclear war using the larger weapons that would more likely produce [[Time to nuclear Armageddon|nuclear Armageddon]].{{cn}}  

* The Trump administration announced the sale of a nuclear reactor to Saudi Arabia.  Critics express concern that this would increase the risk that the Saudis may develop their own nuclear weapons.{{cn}} This in turn is particularly worrying for several reasons.  First, there is [[w:The 28 pages|substantial evidence that leading Saudis including members of the Saudi royal family and employees of the Saudi embassy and consulates in the US]] actively supported the preparations for the [[w:September 11 attacks|September 11 attacks]] of 2001.  Second, [[Winning the War on Terror|the vast majority of Islamic terrorists]] belong to the Wahabbi / Salafist branch of Islam, which is by far the most violent branch of Islam.  Third, the Saudi Government has continued to support {{w|al Qaeda}} at least as recently as 2019.&lt;ref name=SaudiQaeda/&gt;

== Conclusions ==

It seems likely that nuclear proliferation will continue until an international movement has far more success than similar previous efforts in ending it.  The seven recommendations of the Wisconsin Project on Nuclear Arms Control mentioned above may or may not slow nuclear proliferation enough to prevent nuclear Armageddon destroying civilization, dramatically shorten the lives of nearly all humans on earth.  

Might it be possible to energize existing organizations concerned about nuclear proliferation to the point that they have unprecedented success in achieving nearly complete nuclear disarmament and in strengthening international law so the poor, weak and disfranchised have effective nonviolent means for pursuing a redress of grievances?

== Appendix.  Companion R Markdown vignettes == 

Statistical details that make [[w:Reproducibility|the research in article reproducible]] are provided in two R Markdown vignettes on &quot;Forecasting nuclear proliferation&quot; and &quot;GDPs of nuclear-weapon states&quot;:  

* [[Forecasting nuclear proliferation/Simulating nuclear proliferation]]
* [[Forecasting nuclear proliferation/GDPs of nuclear-weapon states]]

== See also == 

* [[Time to extinction of civilization]]
* [[Time to nuclear Armageddon]]

== References == 

* {{cite Q|Q62670082}}&lt;!-- Burnham and Anderson (1998) Model selection and mutimodel inference --&gt;. 

* {{cite Q|Q27500468}}&lt;!-- Toon et al. (2007) Atmospheric effects and societal consequences of regional scale nuclear conflicts and acts of individual nuclear terrorism --&gt;

== Notes ==
{{reflist}}

[[Category:Original research]]
[[Category:Research]]
[[Category:Political science]]
[[Category:Military]]
[[Category:Military Science]]
[[Category:Freedom and abundance]]
[[Category:Reliability]]
[[Category:Reliability engineering]]
[[Category:Survival analysis]]
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:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]], and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].''  

This article (i) describes efforts to model the time between the first test of a nuclear weapon by one nation and the next over the 74 years of history since the first such test by the US,&lt;ref&gt;This is being written on 2020-04-26.  For the purposes of the present analysis, this is considered to be 74 years since the first test of a nuclear weapon on 1945-07-16.&lt;/ref&gt; (ii) forecasts nuclear proliferation over the next 74 years with statistical error bounds quantifying the uncertainty, and (iii) reviews some of the geopolitical questions raised by this effort.  Our modeling effort considers the possibility that the rate of nuclear proliferation may have slowed over time.   

In brief, current international policy seems to imply that nuclear proliferation can be ignored.  The analysis in this article of the statistical and non-statistical evidence suggests that nuclear proliferation is likely to continue unless (a) a nuclear war destroys everyone's ability to make more such weapons for a long time, or (b) an international movement has far more success than similar previous efforts in providing effective nonviolent recourse  for grievances of the poor, weak and disfranchised.  

Statistical details are provided in R Markdown vignettes on “Forecasting nuclear proliferation” and &quot;GDPs of nuclear weapon states&quot; in an appendix, below.  Those vignettes should allow anyone capable of accessing the {{w|free and open-source software}} [[R (programming language)|R]] and [[w:RStudio|RStudio]] to replicate this analysis and modify it in any way they please to check the robustness of the conclusions. 

== The data ==

The “nuclearWeaponStates” dataset&lt;ref&gt;{{cite Q|Q88894684}}&lt;!-- nuclearWeaponStates dataset--&gt;&lt;/ref&gt; in the Ecdat package for R&lt;ref&gt;{{cite Q|Q56452356}}&lt;!-- https://github.com/sbgraves237/Ecdat --&gt;&lt;/ref&gt; was used for this study.  Those data combine information from the “World Nuclear Weapon Stockpile” maintained by Ploughtshares,&lt;ref&gt;{{cite Q|Q63197617}}&lt;!--  World Nuclear Weapon Stockpile compiled by Ploughshares --&gt;&lt;/ref&gt; the Wikipedia article on “[[w:List of states with nuclear weapons|List of states with nuclear weapons]]”, and multiple articles in the {{w|Bulletin of the Atomic Scientists}}.  This includes the five states that officially had nuclear weapons when the United Nations {{w|Treaty on the Non-Proliferation of Nuclear Weapons}} (Non-Proliferation Treaty, NPT) entered into force in 1970 (the US, Russia, the UK, France and China) plus four others that first tested nuclear weapons since (India, Israel, Pakistan, and North Korea).  

There seems to be a fairly broad consensus on the dates of the first tests of 8 of these 9 nuclear weapon states.  Some reports claim that France and Israel had such close collaboration on nuclear weapons development in the late 1950s that the first test of a nuclear weapon by France on 1960-02-13 effectively created two nuclear-weapon states, not one.&lt;ref&gt;{{cite Q|Q88922617}}&lt;!-- The Third Temple's Holy of Holies:  Israel's Nuclear Weapons, tech report by Lt.Col. Warner D. Farr, --&gt;&lt;/ref&gt; The current study used the date of the 1979-09-22 {{w|Vela Incident}} for Israel.  A 2019 report by Professor Avner Cohen, professor at the Middlebury Institute of International Studies, and the Director of the Education Program and Senior Fellow at the James Martin Center for Nonproliferation Studies, said that, “there is a scientific and historical consensus that [the Vela incident] was a nuclear test and that it had to be Israeli”,&lt;ref&gt;{{cite Q|Q88921529}}&lt;!--  U.S. Covered Up an Israeli Nuclear Test in 1979, Foreign Policy Says, article in Haaretz --&gt;&lt;/ref&gt; conducted probably with South Africa.  A robustness analysis could involve simply deleting Israel as a separate nuclear-weapon state.

== Plotting the time between the “first test” by one nuclear-weapon state and the next == 

[[File:NucWeaponStates YrsBetw1stTsts.svg|thumb|Figure 1.  Years between new nuclear-weapon states.   CN = China, FR = France, GB = UK, IL = Israel, IN = India, KP = North Korea, PK = Pakistan, RU = Russia.  NPT = {{w|Treaty on the Non-Proliferation of Nuclear Weapons}} (Non-Proliferation Treaty).  INF = {{w|Intermediate-Range Nuclear Forces Treaty}}.  The US is not on this plot, because it had no predecessors.]]

A plot of times between &quot;first tests&quot; by the world's nuclear-weapon states as of 2020-04-29 suggests that the process of nuclear proliferation has slowed;  see Figure 1.  

This plot also marks the effective dates of both the {{w|Treaty on the Non-Proliferation of Nuclear Weapons}} (Non-Prolireration Treaty, NPT) and the [[w:Intermediate-range Nuclear Forces Treaty|Intermediate-range Nuclear Forces (INF) Treaty]] (1970-03-05 and 1988-06-01, respectively), because of the suggestion that those treaties may have slowed the rate of nuclear proliferation.

A visual analysis of this plot suggests that nuclear proliferation is still alive and well, and neither the NPT nor the INF treaty impacted nuclear proliferation. The image is pretty bad:  There were only 5 nuclear-weapon states when the NPT entered into force in 1970.&lt;ref&gt;{{cite Q|Q91335914}}&lt;!-- Treaty on the Non-Proliferation of Nuclear Weapons --&gt;.  See also {{w|reaty on the Non-Proliferation of Nuclear Weapons}}.&lt;/ref&gt; When US President {{w|George W. Bush}} decried an [[w:Axis of evil|&quot;Axis of evil&quot;]] in his State of the Union message, 2002-01-29,&lt;ref&gt;{{cite Q|Q91337578}}&lt;!--2002 State of the Union Address by US President George W. Bush--&gt;.  See also [[w:Axis of Evil]].&lt;/ref&gt; there were 8.  As this is written 2020-04-21, there are 9.  

Toon et al. (2007) noted that in 2003 another 32 had sufficient fissile material to make nuclear weapons if they wished.  Moreover, those 32 do ''NOT'' include either Turkey nor Saudi Arabia.  On 2019-09-04, Turkish President Erdogan said it was unacceptable for nuclear-armed states to forbid Turkey from acquiring its own nuclear weapons.&lt;ref&gt;{{cite Q|Q91338524}}&lt;!-- Erdogan says it's unacceptable that Turkey can't have nuclear weapons, 2019 Reuters news article by Ece Toksabay--&gt;;  {{cite Q|Q91342138}} &lt;!-- Tom OConnor (2019) “Turkey has U.S. nuclear weapons, Now it says it should be allowed to have some of its own” --&gt;.&lt;/ref&gt;

Similarly, in 2006 ''Forbes'' reported that Saudi Arabia has &quot;a secret underground city and dozens of underground silos for&quot; Pakistani nuclear weapons and missiles.&lt;ref&gt;{{cite Q|Q91342270}}&lt;!-- Forbes:2006:  AFX News Limited: &quot;Saudia Arabia working on secret nuclear program with Pakistan help - report&quot; --&gt;;  see also [[w:Nuclear program of Saudi Arabia]].&lt;/ref&gt; In 2018 the ''Middle East Monitor'' reported that &quot;Israel 'is selling nuclear information' to Saudi Arabia&quot;.&lt;ref&gt;{{cite Q|Q91343477}}&lt;!-- Israel ‘is selling nuclear information’ to Saudi Arabia, per Middle East Monitor --&gt;;  see also [[w:Nuclear program of Saudi Arabia]].&lt;/ref&gt; This is particularly disturbing, because of the substantial evidence that Saudi Arabia may have been and may still be the primary recruiter and funder of Islamic terrorism.&lt;ref&gt;{{cite Q|Q55616039}}&lt;!-- Medea Benjamin (2016) Kingdom of the Unjust: Behind the US-Saudi Connection --&gt;; see also [[Winning the War on Terror]].&lt;/ref&gt;

This analysis suggests that the number of nuclear-weapon states will likely continue to grow until some dramatic break with the past makes further nuclear proliferation either effectively impossible or sufficiently undesirable. 

This article first reviews the data and history on this issue. We then discuss modeling these data as a series of annual Poisson observations of the number of states conducting a first test of a nuclear weapon each year (1 in each of 8 years since 1945; 0 in the others).

A relatively simple model for the inhomogeneity visible in Figure 1 is {{w|Poisson regression}} assuming that log(Poisson mean) is linear in the time since the first test of a nuclear weapon by the US on 1945-07-16.&lt;ref&gt;A vignette on “Forecasting nuclear proliferation” describes fitting such models to the available data in a way that allows anyone able to run the {{w|free and open-source software}} {{w|R (programming language)}} to [[w:Reproducibility#Reproducible research|reproduce the analysis outlined in this article]] and experiment with alternatives:  {{cite Q|Q89780728}}&lt;!-- Forecasting nuclear proliferation--&gt;.&lt;/ref&gt;  This model is plausible to the extent that this trend might represent a growing international awareness of the threat represented by nuclear weapons including a hypothesized increasing reluctance of existing nuclear-weapon states to share their technology. The current process of ratifying the new {{w|Treaty on the Prohibition of Nuclear Weapons}} supports the hypothesis of such a trend, while the lack of universal support for it and the trend visible in Figure 1 clearly indicate that nuclear proliferation is still likely to continue.  We use this model to extend the 74 years of history of nuclear proliferation available as this is being written on 2020-04-21 into predicting another 74 years into the future.

== How did the existing nuclear-weapon states develop this capability? == 

There are, of course, multiple issues in nuclear proliferation:  a new nuclear-weapon state requires at least four distinct things to produce a nuclear weapon: motivation, money, knowledge, and material.  And many if not all of the existing nuclear-weapon states got foreign help, as outlined below and summarized in the accompanying table.  

'''Disclaimer''':  Complete answers to each of these questions for every nuclear-weapon state can never be known with certainty. The literature found by the present authors is summarized in the accompanying table with citations to the literature in the following discussion but should not be considered any more authoritative than the sources cited, some of which may not be adequate to support all the details and the generalizations in the accompanying table.

However, this analysis should be sufficient to support the general conclusions of this article.  

{| class=&quot;wikitable&quot;
|-
! rowspan=&quot;2&quot; | Country 
! rowspan=&quot;2&quot; | Motivation
! rowspan=&quot;2&quot; | Money 
! rowspan=&quot;2&quot; | Knowledge 
! rowspan=&quot;2&quot; | Material 
! colspan=&quot;2&quot; | Foreign Help 
|-
! Who 
! Why 
|-
| US 
| Nazi threat
| self
| own scientists + immigrants, esp. fr. Germany &amp; Italy in collaboration with the UK and Canada.  
| Congo + self 
| GB (incl. Canada)
| Nazi threat
|-
| USSR (RU)
| Hiroshima &amp; Nagasaki bombs + western invasions during WW II, after WW I, and before
| self
| own scientists + espionage in the US &amp; captured Germans
| self 
| US (espionage)
| US scientists wanted to protect USSR
|- 
| UK (GB)
| USSR
| self
| Manhattan Project
| Canada
| colspan=&quot;2&quot; style=&quot;text-align: center;&quot; | ?
|- 
| France (FR)
| USSR + Suez Crisis
| self
| self
| self
| colspan=&quot;2&quot; style=&quot;text-align: center;&quot; | ?
|- 
| China (CN) 
| 1st Taiwan Strait Crisis 1954–1955, the Korean Conflict, etc.
| self
| USSR
| self
| RU 
| US threat 
|- 
| India (IN)
| loss of territory in the China-Himalayan border dispute-1962
| self
| students in UK, US
| Canadian nuc reactor
| colspan=&quot;2&quot; style=&quot;text-align: center;&quot; | ?
|- 
| Israel (IL) 
| hostile neighbors
| self
| self + France
| France + ???
| colspan=&quot;2&quot; style=&quot;text-align: center;&quot; | ?
|- 
| rowspan=&quot;2&quot; | Pakistan (PK)
| rowspan=&quot;2&quot; | Loss of E. Pakistan in 1971
| rowspan=&quot;2&quot; | Saudis + self
| rowspan=&quot;2&quot; | US, maybe China?
| rowspan=&quot;2&quot; | self?
| US || USSR in Afghanistan
|-
| CN || ? 
|- 
| N.Korea (KP)  
| threats fr. US
| self?
| US via Pakistan?
| self?
| PK +? 
| ? 
|}

'''''Table 1.  Where did the existing nuclear-weapon states get the motivation, money, knowledge, and material for their nuclear-weapons program?'''''  

[[File:GDP of nuclear-weapon states (billions of 2019 USD).svg|thumb|Figure 2.  {{w|Gross Domestic Product}} (GDP) of nuclear-weapon states in billions of 2019 US dollars at {{w|Purchasing Power Parity}} (PPP) before (dashed line), during (thick solid line) and after (thinner solid line) their nuclear-weapons program leading to their first test of a nuclear weapon.  (Country codes as with Figure 1.)  The dotted line indicates the total cost of the Manhattan Project that developed the very first nuclear weapon from 1942 to the end of 1945.]]

To help us understand the differences in sizes of the different nuclear-weapon states, Figure 2 plots the evolution of GDP in the different nuclear-weapon states.  The following subsections provide analysis with references behind the summaries in Table 1 and Figure 2.

=== Motivation === 

Virtually any country that feels threatened would like to have some counterweight against aggression by a potential enemy.  

* The US funded the Manhattan project believing that Nazi Germany likely had a similar project.  

* Soviet leaders might have felt a need to defend themselves from nuclear coercion after having been invaded by Nazi Germany only a few years earlier, and having defeated [[w:Allied intervention in the Russian Civil War|foreign invasions from the West and the East after World War I trying to put the Tsar back in power]].&lt;ref&gt;{{cite Q|Q91370284}}&lt;!-- Fogelsong (1995) America's Secret War against Bolshevism: U.S. Intervention in the Russian Civil War, 1917-1920 --&gt;. That doesn't count [[w:|French invasion of Russia|numerous other invasions that are a sordid part of Russian history]], which educated Russians throughout history would likely remember, even if their invaders may not.&lt;/ref&gt;

* The United Kingdom and France felt nuclear threats from the Soviet Union.&lt;ref&gt;The UK and France would have had many reasons to fear the intentions of the USSR during the early period of the {{w|Cold War}}:  The first test of a nuclear weapon by the USSR came just over three months after the end of the 1948-49 {{w|Berlin Blockade}}.  Other aspects of Soviet repression in countries they occupied in Eastern Europe contributed to the failed {{w|Hungarian Revolution of 1956}}.&lt;/ref&gt; France's concern about the Soviets increased [[w:France and weapons of mass destruction#cite note-16|after the US refused to support them during the 1956]] {{w|Suez Crisis}}:  If the US would not support a British-French-Israeli invasion of Egypt, the US might not defend France against a possible Soviet invasion.&lt;ref&gt;{{cite Q|Q91382112}}&lt;!-- Devid Fromkin (2006) Stuck in the Canal, NYT--&gt;.  See also [[w:France and weapons of mass destruction]].&lt;/ref&gt; 

* China reportedly decided to initiate its nuclear weapons program during the [[w:China and weapons of mass destruction#Nuclear weapons|First Taiwan Strait Crisis of 1954-55]],&lt;ref&gt;[[w:China and weapons of mass destruction#Nuclear weapons]];  see also [[w:First Taiwan Strait Crisis]], {{cite Q|Q63874609}}&lt;!--  Morton Halperin (1966) The 1958 Taiwan Straits Crisis: A documentary history --&gt;, and [[w:Daniel Ellsberg]].&lt;/ref&gt; following nuclear threats from the US regarding Korea.&lt;ref&gt;{{cite Q|Q63874136}}&lt;!-- The Atomic Bomb and the First Korean War --&gt;.  See also [[w:Daniel Ellsberg]].&lt;/ref&gt;

* India lost territory to China in the 1962 {{w|Sino-Indian War}}, which reportedly convinced India to abandon a policy of avoiding nuclear weapons.&lt;ref&gt;{{cite Q|Q91391545}}&lt;!-- Bruce Riedel (2012) JFK's Overshadowed Crisis --&gt;.  See also {{w|India and weapons of mass destruction}}.  India and China have continued to have conflicts.  See, for example, the Wikipedia articles on [[w:China-India relations]] and the [[w:2017 China-India border standoff]].&lt;/ref&gt;

* Pakistan's nuclear weapons program began in 1972 in response to the loss of East Pakistan (now Bangledesh) in the 1971 {{w|Bangladesh Liberation War}}.&lt;ref&gt;{{w|Pakistan and weapons of mass destruction}}.  {{w|India-Pakistan relations}} have been marked by frequent conflict since the two nations were born with the dissolution of the British Raj in 1947.  This history might help people understand the need that Pakistani leaders may have felt and still feel for nuclear parity with India, beyond the loss of half their population and 15 percent of their land area in the 1971 Bangladesh Liberation War.&lt;/ref&gt; On November 29, 2016, Moeed Yusuf claimed that the threat of a nuclear war between India and Pakistan was the most serious foreign policy issue facing then-President-elect Trump.&lt;ref&gt;{{cite Q|Q91271615}}&lt;!-- Moeed Yusuf (2016-11-26) “An India-Pakistan Crisis:  Should we care?”, War on the Rocks --&gt;.&lt;/ref&gt; That may have been an overstatement, but the possibilities of a nuclear war between India and Pakistan should not be underestimated.  [[w:Indo-Pakistani wars and conflicts|There have been lethal conflicts between India and Pakistan at least as recent as 2019.]]  If that conflict goes nuclear, it could produce a “nuclear autumn” during which a quarter of humanity not directly impacted by the nuclear war would starve to death, according to simulations by leading climatologists.&lt;ref&gt;Helfand and the references he cited predicted two billion deaths.  With a [[w:world population|world population]] in 2013 of 7.2 billion, less than 8 billion, 2 billion is more than a quarter of humanity.  See &lt;!-- Nuclear famine: two billion people at risk? --&gt;{{cite Q|Q63256454}}.  See also Toon et al. (2007).&lt;/ref&gt;

* Israel has faced potentially hostile neighbors since its declaration of independence in 1948.&lt;ref&gt;{{w|Arab-Israeli conflict}}.  Threats perceived by Israel continue, including the {{w|2018 Gaza border protests}} that have continued at least into 2020.  One might therefore reasonably understand why Israel might feel a need for nuclear weapons and why others might believe that the 1979-09-22 {{w|Vela incident}} was an Israeli nuclear test.&lt;/ref&gt;

* North Korea first tested a nuclear weapon on 2006-10-09,&lt;ref&gt;{{cite Q|Q59596578}}&lt;!-- Jonathan Medalia (2016) Comprehensive Nuclear-Test-Ban Treaty: Background and Current Developments, Congressional Research Service --&gt;;  The US Congressional Research Service in 2016 reported, “The Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO) PrepCom's international monitoring system detected data indicating that North Korea had conducted a nuclear test on January 6, 2016. ... On October 9, 2006, North Korea declared that it had conducted an underground nuclear test.”  For the present purposes, we use the October date declared by North Korea, not the January date reported by CTBTO.  See also {{w|2006 North Korean nuclear test}}.&lt;/ref&gt; less than five years after having been named as part of an &quot;{{w|Axis of evil}}&quot; by US President George W. Bush on 2002-01-29.&lt;ref&gt;{{cite Q|Q91337578}}&lt;!-- 2002 State of the Union Address by US President George W. Bush--&gt;;  see also ''[[w:Axis of evil]]''.&lt;/ref&gt; Chomsky claimed that the relations between the US and North Korea have followed &quot;a kind of tit-for-tat policy.  You make a hostile gesture, and we'll respond with a crazy gesture of our own.  You make an accommodating gesture, and we'll reciprocate in some way.&quot;  He gave several examples including a 1994 agreement that halted North Korean nuclear-weapons development.  &quot;When George W. Bush came into office, North Korea had maybe one [untested] nuclear weapon and verifiably wasn't producing any more.&quot;&lt;ref&gt;{{cite Q|Q86247233}}&lt;!-- Who Rules the World?, 2017 book by Noam Chomsky--&gt;, pp. 131-134. Chomsky includes in this game of tit-for-tat the total destruction of North Korean infrastructure during the Korean War in the early 1950s, including huge dams that controlled the nation's water supply, destroying their crops, and raising the spectre of mass starvation.  {{cite Q|Q91455702}}&lt;!-- Report on the destruction of dikes: Holland 1944-45 and Korea 1953 --&gt; noted that German General Syss-Inquart ordered similar destruction of dikes in Holland in 1945, which condemned many Dutch civilians to death by starvation. For that crime Syss-Inquart became one of only 24 of the people convicted at the Nurenberg war crimes trial to have been sentencted to death.  Chomsky noted that this is &quot;not in our memory bank, but it's in theirs.&quot;&lt;/ref&gt;

All this suggests that it will be difficult to reduce the threat of nuclear proliferation and nuclear war without somehow changing the nature of international relations so weaker countries have less to fear from the demands of stronger countries.

=== Money ===

It's no accident that most of the world's nuclear-weapon states are large countries with substantial populations and economies. That's not true of Israel with only roughly 9 million people nor North Korea with roughly 26 million people in 2018.  France and the UK have only about 67 and 68 million people, but they are also among the world leaders in the size of their economies.  

Pakistan is a relatively poor country.  It reportedly received financial assistance from Saudi Arabia for its nuclear program.&lt;ref&gt;{{cite Q|Q84288832}}&lt;!-- Saudi Arabia: Nervously Watching Pakistan --&gt;.&lt;/ref&gt;   

Another reason for a possible decline in the rate of nuclear proliferation apparent in Figure 1 is the fact that among nuclear-weapon states, those with higher GDPs tended to acquire this capability earlier, as is evident in Figure 2.

=== Knowledge ===

In 1976, {{w|John Aristotle Phillips}}, an &quot;underachieving&quot; undergraduate at Princeton University, &quot;designed a nuclear weapon using publicly available books and papers.&quot;&lt;ref&gt;{{cite Q|Q91459264}}&lt;!-- Student Designs Nuclear Bomb (1976-10-09) Spokane Daily Chronicle--&gt;. See also [[w:John Aristotle Phillips]].&lt;/ref&gt; Nuclear weapons experts disagreed on whether the design would have worked.  Whether Phillips' design would have worked or not, it should be clear that the continuing progress in human understanding of {{w|nuclear physics}} inevitably makes it easier for people interested in making such weapons to acquire the knowledge of how to do so.  

Before that, the nuclear age arguably began with the 1896 discovery of radioactivity by the French scientist Henri Becquerel.  It was further developed by Pierre and Marie Curie in France, Ernest Rutherford in England, and others, especially in France, England and Germany.&lt;ref&gt;{{w|Nuclear physics}}.&lt;/ref&gt; In 1933 after Adolf Hitler came to power in Germany, {{w|Leo Szilard}} moved from Germany to England. The next year he patented the idea of a nuclear fission reactor.  Other leading nuclear scientists similarly left Germany and Italy for the UK and the US.  After World War II began, the famous {{w|Manhattan Project}} became a joint British-American project, which produced the very first test of a nuclear weapon.&lt;ref&gt;{{w|History of nuclear weapons}}.&lt;/ref&gt; 

After Soviet premier {{w|Joseph Stalin}} learned of the atomic bombings of Hiroshima and Nagasaki, the USSR (now Russia) increased the funding for their nuclear-weapons program.  That program was helped by intelligence gathering about the German nuclear weapon project and the American Manhattan Project.&lt;ref&gt;{{cite Q|Q91461780}}&lt;!-- Espionage and the Manhattan Project (1940-1945), Office of Scientific and Technical Information, US Department of Energy --&gt;.  See also {{w|Soviet atomic bomb project}}.&lt;/ref&gt;

The UK's nuclear-weapons program was built in part on their wartime participation in the Manhattan Project, as noted above.  

France was among the leaders in nuclear research until World War II.  They still had people with the expertise needed after the 1956 {{w|Suez Crisis}} convinced them they needed to build nuclear bombs, as noted above.&lt;ref&gt;See also {{w|History of nuclear weapons}}.&lt;/ref&gt;

China got some help from the Soviet Union during the initial phases of their nuclear program.&lt;ref&gt;{{w|China and weapons of mass destruction}}.&lt;/ref&gt;

The first country to get nuclear weapons after the Non-Proliferation Treaty was India.  Their Atomic Energy Commission was founded in 1948, chaired by {{w|Homi J. Bhabha}}.  He had published important research in nuclear physics while a graduate student in England in the 1930s, working with some of the leading nuclear physicists of that day.&lt;ref&gt;[[w: Homi J. Bhabha]];  see also [[w:Timeline of nuclear weapons development]].&lt;/ref&gt;

Meanwhile, Israel's nuclear weapons program initially included sending students abroad to study under leading physicists like Enrico Fermi at the University of Chicago.  It also included extensive collaboration with the French nuclear-weapons program.&lt;ref&gt;[[w:Nuclear weapons and Israel]].  See also [https://www.wisconsinproject.org/israels-nuclear-weapon-capability-an-overview/ &quot;Israel’s Nuclear Weapon Capability: An Overview&quot;], July 1, 1996, by the Wisconsin Project on Nuclear Arms Control.&lt;/ref&gt;

Pakistan got &quot;dual use&quot; production technology and complete nuclear-capable delivery systems from both the US and China.&lt;ref&gt;For Chinese help to Pakistan, see {{cite Q|Q95917195}}&lt;!--  Gradual Signs of Change: Proliferation to and from China over Four Decades --&gt;.&lt;/ref&gt; Pakistan got secret help from the US in the 1980s in violation of US law to secure Pakistani cooperation with US support for anti-Soviet resistance in Afghanistan.&lt;ref&gt;{{cite Q|Q91463994}}&lt;!-- New Documents Spotlight Reagan-era Tensions over Pakistani Nuclear Program, research report by William Burr, Wilson Center --&gt;. {{cite Q|Q91464530}}&lt;!-- Pakistan's Illegal Nuclear Procurement Exposed in 1987: Arrest of Arshed Pervez Sparked Reagan Administration Debate over Sanctions, National Security Archive Electronic Briefing Book No. 446 --&gt;. See also [[w:Pakistan and weapons of mass destruction]].&lt;/ref&gt; (In 1995 the Wisconsin Center on Nuclear Arms Control reported that Pakistan’s most reliable nuclear delivery platforms were French-made Mirage fighters,&lt;ref&gt;{{cite Q|Q95919096}}&lt;!-- Pakistan: American, Chinese or French Planes Would Deliver its Bomb, Wisconsin Project on Nuclear Arms Control --&gt;.&lt;/ref&gt; though they also had US-made F-16s they could modify to carry those weapons.)  

{{w|Abdul Qadeer Khan}}, a leader in Pakistan's nuclear weapons program, has also faced multiple allegations of being one of the world's leading nuclear proliferators in operating a black market in nuclear weapons technology.  North Korea, Iran and other countries have allegedly received help from Pakistan for their nuclear weapons programs with at least some of it coming via A. Q. Khan's black market dealings.&lt;ref&gt;A summary of this appears in [https://www.wisconsinproject.org/pakistan-nuclear-milestones-1955-2009/ &quot;Pakistan Nuclear Milestones, 1955-2009&quot;] by the {{w|Wisconsin Project on Nuclear Arms Control}}.  See also citations on this in the Wikipedia article on [[w:Abdul Qadeer Khan]].&lt;/ref&gt; Some of this technology was reportedly obtained from the US in the 1980s with the complicity of US government officials who wanted Pakistan's help for groups in Afghanistan fighting the Soviets.&lt;ref&gt;E.g., {{cite Q|Q88306915}}&lt;!-- Lyndsey Layton (7 July 2007), &quot;Whistle-Blower's Fight For Pension Drags On&quot;, The Washington Post --&gt;, and [[w:Richard Barlow (intelligence analyst)|Richard Barlow]].&lt;/ref&gt; 

{{w|Vikram Sood}}, a former head of India's foreign intelligence agency, said, &quot;America fails the IQ test&quot; in discussing A. Q. Khan's nuclear black market, adding that Pakistan ''may'' have given nuclear-weapons technology to al Qaeda &quot;just weeks prior to September 11, 2001.&quot;&lt;ref&gt;{{cite Q|Q88310866}}&lt;!-- America fails the IQ test--&gt;&lt;/ref&gt; It may not be wise to accept Sood's claim at face value, given the long-standing hostility between India and Pakistan. In April 2002 Milhollin, Founder and then Executive Director of the Wisconsin Project on Nuclear Arms Control, said that Al Qaeda &quot;is interested in getting weapons of mass destruction, [and if it] can organize a 19-person group to fly airliners into buildings, it can smuggle a nuclear weapon across a border.&quot;&lt;ref&gt;{{cite Q|Q95987528}}&lt;!-- Use of Export Controls to Stop Proliferation --&gt;.&lt;/ref&gt; In 2005 Robert Gallucci, a leading researcher and expert on nuclear proliferation who served in high level positions in the Reagan, G. H. W. Bush and Clinton administrations because of this expertise, wrote that there was an unacceptably high probability &quot;that Al Qaeda or one of its affiliates will detonate a nuclear weapon in a US city ... . The loss of life will be measured ... in the hundreds of thousands. ... Consider the more likely scenarios ... .  An Al Qaeda cell ... purchases 50 or so kilograms of highly enriched uranium.  Today, the sellers might be Pakistan or Russia;  tomorrow they might be North Korea or Iran.  ... Another scenario ... involves the acquisition ... of a completed nuclear weapon.&quot;&lt;ref&gt;Gallucci's estimate of the probability of a nuclear attack by a terrorist group has declined substantially since 2005.  Back then, he wrote that a terrorist attack with a nuclear weapon in the next five to ten years &quot;is more likely than not&quot;.  In a private communication on June 4, 2020, he wrote, &quot;I was wrong in my estimate [that such an attack was more likely than not], and glad that I was. I don't understand AQ to be the threat now that it was fifteen years ago, but my concern continues that it is principally the unavailability of fissile material that prevents a terrorist from constructing an improvised nuclear device.&quot; The quote from 2005 is available in {{cite Q|Q96062427}}&lt;!-- Averting Nuclear Catastrophe: Contemplating Extreme Responses to U.S. Vulnerability, Harvard International Review, 2005, pp. 84, 83 --&gt;.  Essentially this same quote appears in a longer article by the same name:  {{cite Q|Q29395474}}&lt;!--Averting Nuclear Catastrophe: Contemplating Extreme Responses to U.S. Vulnerability, Annals of the American Academy of Political and Social Science, 2006--&gt;.&lt;/ref&gt;
  
And the US is helping Saudi Arabia obtain nuclear power, in spite of (a) the evidence that [[w:The 28 pages|the Saudi government including members of the Saudi royal family were involved at least as early as 1999 in preparations for the suicide mass murders of September 11, 2001]],&lt;ref&gt;{{cite Q|Q1702537}}&lt;!-- Joint inquiry into intelligence community activities before and after the terrorist attacks of September 11, 2001 --&gt;.  See also {{w|The 28 pages}}, which were redacted from the official report published 2003-01-29 and declassified in July 2016 by then-President Obama.&lt;/ref&gt; and (b) their [[w:Saudi Arabian-led intervention in Yemen|on-going support for Al Qaeda in Yemen, reported as recently as 2018]].&lt;ref name=SaudiQaeda&gt;{{cite Q|Q61890713}}&lt;!-- AP Investigation: US allies, al-Qaida battle rebels in Yemen--&gt;.&lt;/ref&gt;

=== Material === 

Reportedly the most difficult part of making nuclear weapons today is obtaining sufficient fissile material.  Toon et al. (2007) said, &quot;Thirteen countries operate plutonium and/or uranium enrichment facilities, including Iran&quot;, but Iran did not have sufficient fissile material in 2003 to make a nuclear weapon.  Another 20 were estimated to have had sufficient stockpiles of fissile material acquired elsewhere to make nuclear weapons.  They concluded that 32 (being 13 minus 1 plus 20) additional countries have sufficient fissile material to make nuclear weapons if they want.&lt;ref&gt;pp. 1975, 1977.  The 32 countries they identified included 12 of the 13 that &quot;operate plutonium and/or uranium enrichment facilities&quot;, excepting Iran as noted.  The other 20 countries acquired stockpiles elsewhere. In addition to the 32 with sufficient fissile material to make a nuclear weapon, Egypt, Iraq and the former Yugoslavia were listed as having abandoned a nuclear-weapons program.&lt;/ref&gt;

Toon et al. (2007) also said, &quot;In 1992 the International Atomic Energy Agency safeguarded less than 1% of the world’s HEU [Highly Enriched Uranium] and only about 35% of the world inventory of Pu [Plutonium] ... .  Today [in 2007] a similarly small fraction is safeguarded.&quot; 

HEU is obtained by separating &lt;sup&gt;235&lt;/sup&gt;U, which is only 0.72 percent of naturally occurring uranium.&lt;ref&gt;{{cite Q|Q91488549}}&lt;!-- Weapons of Mass Destruction (WMD):  Uranium Isotopes --&gt;.&lt;/ref&gt; Weapons-grade uranium has at least 85 percent &lt;sup&gt;235&lt;/sup&gt;U.&lt;ref&gt;See the section on “Highly enriched uranium (HEU)” in the Wikipedia article on [[w:Enriched uranium]].&lt;/ref&gt; Thus, at least 0.85/0.0072 = 118 kg of naturally occurring uranium are required to obtain 1 kg that is weapons-grade.  Toon et al. (2007) estimated that 25 kg of HEU would be used on average for each &lt;sup&gt;235&lt;/sup&gt;U-based nuclear weapon.  Plutonium, by contrast, is a byproduct of energy production in standard &lt;sup&gt;238&lt;/sup&gt;U nuclear reactors. 

Much of the uranium for the very first test of a nuclear weapon by the US came from the Congo,&lt;ref name='Ures'&gt;[[w:Manhattan project]].&lt;/ref&gt; but domestic sources provided most of the uranium for later US nuclear-weapons production.&lt;ref&gt;[[w:List of countries by uranium reserves]].&lt;/ref&gt; The Soviet Union (USSR, now Russia) also seems to have had adequate domestic sources for its nuclear-weapons program, especially including Kazakhstan, which was part of the USSR until 1990;  Kazakhstan has historically been the third largest source of uranium worldwide after Canada and the US.&lt;ref name='Ures'/&gt; The UK presumably got most of its uranium from Canada.  

The French nuclear-weapons program seems to have been built primarily on plutonium.&lt;ref&gt;{{w|France and weapons of mass destruction}}.  See also Table 2 in Toon et al. (2007), which claims that in 2003, France had enough fissile material for roughly 24,000 plutonium bombs and 1,350 &lt;sup&gt;235&lt;/sup&gt;U bombs.&lt;/ref&gt; This required them to first build standard &lt;sup&gt;238&lt;/sup&gt;U nuclear reactors to make the plutonium.  Then they didn't need nearly as much uranium to sustain their program.  

China has reportedly had sufficient domestic reserves of uranium to support its own needs,&lt;ref name='Ures'/&gt; even exporting some to the USSR in the 1950s in exchange for other assistance with their nuclear defense program.&lt;ref&gt;[[w:China and weapons of mass destruction]].&lt;/ref&gt; 

India's nuclear weapons program seems to have been entirely (or almost entirely) based on plutonium.&lt;ref&gt;[[w:India and weapons of mass destruction]];  see also Toon et al. (2007) and [[w:List of countries by uranium reserves]].&lt;/ref&gt; 

Israel seems not to have had sufficient uranium deposits to meet its own needs.  Instead, they purchased some from France until France ended their nuclear-weapons collaboration with Israel in the 1960s. To minimize the amount of uranium needed, nearly all Israeli nuclear weapons seem to be plutonium bombs.&lt;ref&gt;Toon et al. (2007).&lt;/ref&gt;

It's not clear where Pakistan got most of its uranium: Its reserves in 2015 were estimated at zero, and its historical production to that point was relatively low.&lt;ref name='Ures'/&gt; By comparison with the first seven nuclear-weapon states, it's not clear where Pakistan might have gotten enough uranium to produce 83 plutonium bombs and 44 uranium bombs, as estimated by Toon et al. (2007, Table 2, p. 1976.)  As previously noted, the US helped the Pakistani nuclear-weapons program in the 1980s and accused China of providing similar assistance, a charge that China has repeatedly and vigorously denied.  China has provided civilian nuclear reactors, which could help produce plutonium but not &lt;sup&gt;235&lt;/sup&gt;U.&lt;ref&gt;[[w:Pakistan and weapons of mass destruction#Alleged foreign co-operation]].&lt;/ref&gt;

According to the Federation of American Scientists, &quot;North Korea maintains uranium mines with an estimated four million tons of exploitable high-quality uranium ore ... that ... contains approximately 0.8% extractable uranium.&quot;&lt;ref&gt;{{cite Q|Q91520731}}&lt;!--DPRK: Nuclear Weapons Program per the Federation of American Scientists--&gt;.  See also [[w:North Korea and weapons of mass destruction]].&lt;/ref&gt; If that's accurate, processing all that would produce 4,000,000 times 0.008 = 32,000 tons of pure natural uranium, which should be enough to produce the weapons they have today.  

=== Conclusions regarding motivation, money, knowledge, and material ===

1. There seems to be no shortage of motivations for other countries to acquire nuclear weapons.  The leaders of the Soviet Union had personal memories of being invaded not only by Germany during World War II but also by the US and others after World War I.  The UK had reason to fear the Soviets in their occupation of Eastern Europe.  The French decided after Suez they couldn't trust the US to defend them.  China had been forced to yield to nuclear threats before starting their nuclear program, as did India, Pakistan and North Korea.  Israel has fought multiple wars since their independence in 1948.  
  
2. The knowledge and material required to make such weapons in a relatively short order are also fairly widely available, even without the documented willingness of current nuclear powers to secretly help other countries acquire such weapons in some cases.&lt;ref&gt;In addition to the 32 currently non-nuclear-weapon states with &quot;sufficient fissile material to make nuclear weapons if they wished&quot;, per Toon et al. (2007), the inspector general of the US Department of Energy concluded in 2009 (in its most recent public accounting) that enough highly enriched uranium was missing from US inventories to make at least five nuclear bombs comparable to those that destroyed substantial portions of Hiroshima and Nagasaki in 1945.  The issue of missing fissile material is likely much larger than what was reported missing from US inventories, because substantially more weapons-grade material may be missing in other countries, especially Russia, as noted by {{cite Q|Q91521732}}&lt;!-- Plutonium is missing, but the government says nothing --&gt;.&lt;/ref&gt;
  
3. Unless there is some fundamental change in the structure of international relations, it seems unwise to assume that there will not be more nuclear-weapon states in the future, with the time to the next &quot;first test&quot; of a nuclear weapon following a probability distribution consistent with the previous times between &quot;first tests&quot; of nuclear weapons by the current nuclear-weapon states.

== Distribution of the time between Poisson “first tests” ==

Possibly the simplest model for something like the time between &quot;first tests&quot; in an application like this is to assume they come from one {{w|exponential distribution}} with 8 observed times between the 9 current nuclear-weapon states plus one [[w:Censoring (statistics)|censored observation]] of the time between the most recent one and a presumed next one. This simple theory tells us that the maximum likelihood estimate of the mean time between such &quot;first tests&quot; is the total time from the US &quot;Trinity&quot; test to the present, 74.8 years, divided by the number of new nuclear-weapon states, 8, not counting the first, which had no predecessors.  Conclusion:  Mean time between &quot;first tests&quot; = 9.3 years.&lt;ref&gt;For precursors to the current study that involve censored estimation of time to a nuclear war, see [[Time to extinction of civilization]] and [[Time to nuclear Armageddon]].&lt;/ref&gt;

However, Figure 1 suggests that the time between &quot;first tests&quot; of succeeding nuclear-weapon states is increasing.  The decreasing hazard suggested by this figure requires mathematics that are not as easy as the censored data estimation as just described. 

[[File:NucWeaponStates logYrsBetw1stTsts.svg|thumb|Figure 3.  Semilog plot of the years between new nuclear-weapon states.  (Country codes as with Figure 1.)]]

To understand the current data better, we redo Figure 1 with a log scale on the y axis in Figure 3.   

Figures 1 and 3 seem consistent with the following:

* If the mean time between &quot;first tests&quot; is increasing over time, as suggested by Figures 1 and 3, then the distribution cannot be exponential, because that requires a constant [[w:Survival analysis#Hazard function and cumulative hazard function|hazard rate]].&lt;ref&gt;For the exponential distribution, &lt;math&gt;h(t) = (-d/dt \log S(t)) = \lambda&lt;/math&gt;, writing the exponential survival function as &lt;math&gt;S(t) = \exp(-\lambda t)&lt;/math&gt;.&lt;/ref&gt;

* Even though nuclear proliferation has been slowing since 1950, it seems not to have slowed fast enough to support the assumption that nuclear proliferation can be ignored, which seems to be implied by current international policy.  It could ''accelerate'' in the future if more states began to perceive greater threats from other nations.  

* Fortunately we can simplify this modeling problem by using the famous duality between exponential time between events and a Poisson distribution for numbers of events in specific intervals of time.  By modeling Poisson counts of &quot;first tests&quot; each year, we can use techniques for Poisson regression for models suggested by Figure 3.  The simplest such model might consider log(Poisson mean numbers of &quot;first tests&quot; each year) to be linear in the time since the first test of a nuclear weapon (code-named [[w:Trinity (nuclear test)|&quot;Trinity&quot;]]).&lt;ref&gt;{{cite Q|Q7749726}}&lt;!-- Richard Rhodes (1986) The Making of the Atomic Bomb --&gt;.  See also [[w:Trinity (nuclear test)]].&lt;/ref&gt;  

* The image in Figure 3 suggests the time between “first tests” by new nuclear-weapon states may be increasing, but not necessarily liearly.  Easily tested alternatives to linearity could be second, third and fourth powers of the &quot;timeSinceTrinity&quot;.&lt;ref&gt;One might also consider a model with the log(Poisson mean) behaving like a [[w:Wiener process|&quot;Wiener process&quot; (also called a &quot;Brownian motion&quot;)]].  This stochastic formulation would mean that the variance of the increments in log(hazard) between &quot;first tests&quot; is proportional to the elapsed time.  See {{cite Q|Q91547149}}&lt;!-- Wolfram: Wiener Process--&gt; and [[w:Wiener process]].  The “bssm” package for R should provide a reasonable framework for modeling this;  see {{cite Q|Q91626942}}&lt;!-- bssm: Bayesian Inference of Non-Linear and Non-Gaussian State Space --&gt;.  However, this author's efforts to use this package for this purpose have so far produced unsatisfactory results.  More time understanding the software might produce better results but not necessarily enough better to justify the effort that might be required.&lt;/ref&gt;

We used Poisson regression to model this as a series of the number of events each year.&lt;ref&gt;We could have used one observation each month, week, or day.  Such a change might give us slightly better answers while possibly increasing the compute time more than it's worth.&lt;/ref&gt;

== Parameter estimation == 

For modeling and parameter estimation, we model the number of “first tests” of a new nuclear-weapon state each year (1 in 8 years, 0 in the remaining 66 years between 1945 and 2019) with log(Poisson mean number of “first tests” each year) as polynomials in “timeSinceTrinity” = the time in years since the [[w:Trinity (nuclear test)|Trinity test by the US]], 1945-07-16.  The standard {{w|p-value}} for the {{w|Wald test}} of the linear model was 0.21 -- ''not'' statistically significant.  

{{w|George Box}} famously said that, [[w:All models are wrong|''&quot;All models are wrong, but some are useful.&quot;'']].&lt;ref&gt;{{cite Q|Q91658340}}&lt;!-- Empirical Model-Building and Response Surfaces --&gt;.&lt;/ref&gt;

Burnham and Anderson (1998) and others claim that better predictions can generally be obtained using Bayesian Model Averaging (BMA).&lt;ref&gt;See also {{cite Q|Q91670340}}&lt;!-- Bayesian model selection in social research, Adrian Raftery 1995 --&gt; and {{cite Q|Q62568358}}&lt;!-- Model selection and model averaging, Claeskens and Hjort, 2008 --&gt;.&lt;/ref&gt; In this case, we have two models:  log(Poisson mean) being constant or linear in “timeSinceTrinity”.  It is standard in the BMA literature to assume a priori an approximate uniform distribution over all models considered with a penalty for estimating each additional parameter to correct for the tendency of the models to overfit the data.  With these standard assumptions, this comparison of these two models estimated a 21 percent posterior posterior probability for the model linear in  “timeSinceTrinity”, leaving 79 percent probability for the model with a constant Poisson mean.  

[[File:NucWeaponStates BMAyrsBetw1stTsts.svg|thumb|Figure 4.  BMA constant-linear and quartic fits to time between new nuclear-weapon states.  (Country codes as with Figure 1.)]]

We also experimented with fitting up to quartic models in “timeSinceTrinity”.&lt;ref&gt;{{cite Q|Q91674106}}&lt;!-- BMA: Bayesian Model Averaging package for R --&gt;.  The algorithm used for this retained only the intercept and the coefficient of the highest power in each order.  Models like &lt;math&gt;b_0 + b_1 x + b_2 x^2&lt;/math&gt; with &lt;math&gt;b_1 \ne 0&lt;/math&gt; were considered but had a posterior probability so low they were not retained in the final mixture of models.  The quartic mixture retained only &lt;math&gt;b_0&lt;/math&gt; (constant), &lt;math&gt;b_0 + b_1 x&lt;/math&gt; (linear), &lt;math&gt;b_0 + b_2 x^2&lt;/math&gt; (quadratic), &lt;math&gt;b_0 + b_3 x^3&lt;/math&gt; (cubic), and &lt;math&gt;b_0 + b_4 x^4&lt;/math&gt; (quartic) with posterior probabilities 49.59, 13.24, 13.21, 12.66, and 12.31 percents, respectively.&lt;/ref&gt; These prediction lines were added to Figure 3 to produce Figure 4.  

Comparing predictions between the constant-linear and constant-quartic mixtures might help us understand better the limits of what we can learn from the available data.  A visual analysis of the right (quartic mixture) panel in Figure 4 makes one wonder if the quartic, cubic and quadratic fits are really almost as good as the linear, as suggested by minor differences in the posterior probabilities estimated by the algorithm used.  

However, the forecasts of nuclear proliferation will be dominated by the constant component of the BMA mixture.  Its posterior probability is 79 percent for the constant-linear mixture and 48.59 percent for the quartic mixture.  That means that the median line and all the lower quantiles of all simulated futures based on these models would be dominated by that constant term.  

Moreover, the quartic, cubic and quintic lines in the right (quartic mixture) panel of Figure 4 do not look nearly as plausible, at least to the present author, as the constant and linear lines.&lt;ref&gt;Recall that the estimation methodology here is Poisson regression, not ordinary least squares.&lt;/ref&gt; That, in turn, suggests that the constant linear mixture may be more plausible than the quartic mixture.  

We then used [[w:Monte Carlo method|Monte Carlo simulations]] with 5,000 random samples to compute central 60 and 80 percent confidence limits for the mean plus 80 percent prediction, and (0.8, 0.8) tolerance limits for future nuclear proliferation, as discussed in the next three sections of this article.&lt;ref name='cipiti'&gt;”{{w|Confidence intervals}}&quot; bound the predicted mean number of nuclear-weapon states for each future year considered.  Central 80 percent “{{w|prediction intervals}}&quot; are limits that include the central 80 percent of distribution of the number of nuclear-weapon states.  They add the uncertainty in the modeled Poisson process to the uncertainty of estimating the mean of that process for each future year considered.  We will also compute (0.8, 0.8) &quot;[[w:tolerance interval#Relation to other intervals|tolerance  intervals]]&quot;; &lt;math&gt;(p, 1-\alpha)&lt;/math&gt; tolerance intervals have a probability of &lt;math&gt;(1-\alpha)&lt;/math&gt; of containing a proportion of at least &lt;math&gt;p&lt;/math&gt; of all future observations.&lt;/ref&gt;

== Confidence limits == 

[[File:NucWeaponStates FcstMeanTimeBetw1stTsts.svg|thumb|Figure 5.  Confidence limits for the mean time between “first tests,” past and future per the constant-linear mixture.  (Country codes as with Figure 1.)]]

To get confidence limits, we simulated 5,000 Poisson mean numbers of &quot;first tests&quot; by new nuclear-weapon states for each of the 74 years used in the two BMA fits and another 74 years beyond.  These simulations were later used to compute confidence limits for the model estimates of the Poisson mean and prediction and tolerance limits for the actual number of nuclear-weapon states.&lt;ref name='cipiti'/&gt; First, however, we inverted the simulated Poisson means to get simulated exponential times, then summarized them to get simulated mean, median, and 60 and 80 percent confidence limits of the mean time to the next new nuclear weapon state.  We then added those simulation summary statistics from the constant-linear model in Figure 3 to produce Figure 5.  

The fairly flat shape of the median and lower 10 and 20 percent lines in Figure 5 seem consistent with a model that is a sum of a mixture of log-normal distributions with the dominant component having a posterior probability of either 79 or 48.59 percent and a constant mean, as noted in Figure 4.  The substantial curvature of the solid line forecast looks hopeful, with a mean of simulated means for the constant-linear mixture being almost 200 years between successive &quot;first tests&quot; by new nuclear-weapon states by the end of the forecasted period, 2093.  

The fact that the mean of the simulations exceeds the upper confidence limit for 2093 seems odd but can be explained by noting that this is a mixture of log-normal distributions, and the mean of a log-normal can exceed any quantile of its distribution if the standard deviation is sufficiently large.&lt;ref&gt;This follows, because quantile &lt;math&gt;q&lt;/math&gt; of a log-normal is &lt;math&gt;\exp[\mu+\sigma\Phi^{-1}(q)]&lt;/math&gt; and the mean is &lt;math&gt;\exp[\mu+\sigma^2/2]&lt;/math&gt;, so the mean exceeds quantile &lt;math&gt;q&lt;/math&gt; whenever &lt;math&gt;\sigma\Phi^{-1}(q) &lt; \sigma^2/2&lt;/math&gt;, i.e., when &lt;math&gt;\Phi^{-1}(q) &lt; \sigma/2&lt;/math&gt;.&lt;/ref&gt;

[[File:NucWeaponStates QuarticFcstMeanTimeBetw1stTsts.svg|thumb|Figure 6:  Confidence limits for the mean time between “first tests,” past and future per the quartic mixture.  (Country codes as with Figure 1.)]]

Note further that the distribution for each year in Figure 5 is a mixture of log-normal distributions, which means that their reciprocals, the mean numbers of &quot;first tests&quot; each year, will also be a mixture of log-normals with the same standard deviations on the log scale.  This standard deviation is larger the farther we extrapolate into the future.  

The increase over time in the ''mean'' time between &quot;first tests&quot; in Figures 5 and 6 suggests a desirable decrease in the rate of nuclear proliferation.  

However, we are more concerned with the ''shorter'' times between &quot;first tests&quot;, and they seem all too probable, as we shall see when we simulate and compute their cumulative sums.  To do that, we append these simulated predictions to a plot of the evolution of the number of nuclear-weapon states through the historical period.&lt;ref&gt;In these simulations, we assume a zero probability of a nuclear power giving up their nuclear weapons, even though [[w:South Africa and weapons of mass destruction|South Africa reportedly discontinued their nuclear weapons program in 1989]], prior to its [[w:South_Africa#End of apartheid|first universal elections in 1994]].  We could potentially add South Africa to our dataset of nuclear weapon states with the same date as Israel, then model the distribution of the time to when a nuclear-weapon state gives up its nuclear weapons using an exponential distribution.  For that, we have one observed time and eight such times that are censored.  Standard theory in that case says that the maximum likelihood estimate of the mean time to relinquishing nuclear weapons assuming an exponential distribution is the sum of all the times, censored or observed, divided by the number of times observed, not including the censored times in the denominator. For purposes of illustration, we will assume that South Africa dismantled its nuclear weapons 1989-12-31, though a report of an inspection by the International Atomic Energy Agency dated 1994-08-19 said they had dismantled six nuclear weapons and were still working to dismantle one more.  Based on this, the mean lifetime of a nuclear-weapon state can be estimated at 493 years.  We could potentially add this to the current modeling effort, but it would not likely change the answers enough to justify the additional effort.&lt;/ref&gt;

[[File:NucWeaponStates nucProlifPred.svg|thumb|Figure 7: Confidence limits on the number of nuclear-weapon states, past and predicted mean;  BMA constant-linear model on the left; quartic fit on the right. ]]

These numbers are plotted in Figure 7 for both BMA models considered.  The slope of the median lines are steeper than the recent history, but the statistical evidence does not support the naive interpretation of a slowing in nuclear proliferation that one might get from considering only the most recent data.  

Comparing the forecasts between the constant-linear and quartic BMA mixtures shows that the higher order quartic mixture widens the confidence limits, making the 20th percentile essentially flat with almost no additional nuclear proliferation, while the mean quickly escapes the upper limit.  That sharply rising mean suggests that less than 10 percent of the simulations predict nuclear arms races that involve many nation states and many more non-state armed groups.  These outcomes are not likely, but the probabilities of such outcomes seem too large to be dismissed without further consideration, especially when gambling with the future of civilization.  (Replications of these simulations with different sets of random numbers confirmed the stability of the images in Figure 7.)  

Ignoring the simulations of uncontrolled nuclear arms races, the median lines in Figure 7 predict between 16.3 and 14.5 at the end of the current simulated period, 2093, adding either 7.3 and 5.5 (for the constant-linear and quartic mixtures, respectively) to the current 9 nuclear-weapon states.  Those ''median'' numbers are a little less than double the number of nuclear-weapon states today.  

We extend this analysis by adding prediction intervals to these plots.

== Prediction limits == 

[[File:NucWeaponStates nucProlifPredInt.svg|thumb|Figure 8: Prediction limits on the number of nuclear-weapon states, past and predicted;  BMA constant-linear model on the left; quartic fit on the right.]]

The simplest bounds on the future are {{w|prediction interval}}s, which combine the statistical uncertainty in the estimates of mean numbers of nuclear-weapon states with the random variability in the outcomes.  We simulated 80 percent equal-tailed prediction limits and added them to Figure 7 to produce Figure 8.  

For both Bayesian mixture models, the most likely scenarios, especially the median line and the space between the 60 percent confidence limits, predict a continuation of nuclear proliferation.  It's difficult to imagine how that could continue without also substantively increasing the risk of nuclear war and therefore also of the extinction of civilization.  

[[File:NucWeaponStates nucProbs.svg|thumb|Figure 9:  Probabilities of the time to the next 5 new nuclear-weapon states using the constant-linear and quartic BMA models (left and right panels, respectively).]]

We can also summarize the simulations to estimate the probabilities of having 1, 2, 3, 4, and 5 new nuclear weapon states for each year in the prediction period between 2020 and 2093 in Figure 9.  This is another way of evaluating the sensibility of pretending there will be no further nuclear proliferation:  Not likely.  

Ninety-four percent of the simulations per the constant-linear model had at least one more nuclear-weapon state by 2093 and a 40 percent chance of at least 1 by 2025.  The quartic model predicts a 73 percent chance of at least one more nuclear-weapon state by 2093 and a 29 percent chance of at least one by 2024.  

The conclusions from both models include the following:  

:'''''The current structure of international relations'''''
:'''''seems to threaten the extinction of civilization.'''''

To better quantify the uncertainty in modeling, we can also construct tolerance intervals for the time to the next new nuclear-weapon state.

== Tolerance limits == 

[[File:NucWeaponStates nucProlifTolInt.svg|thumb|Figure 10:  Number of nuclear-weapon states with prediction and tolerance limits;  BMA constant-linear model on the left; quartic fit on the right.]]

We want to add statistical tolerance limits to Figure 8 in addition to the prediction limits.  To do this, we add Poisson simulations to the 80 percent confidence limits in Figure 7 rather than adding Poisson simulations to ''all the individual'' simulations summarized in Figure 7 to produce Figure 8.  The results appear in Figure 10.  

The upper limit lines in Figure 10 are higher than those in Figure 8.  It gives us a bit more humility regarding the value of current knowledge.  However, the difference is not enough to substantively alter our conclusions, namely that nuclear proliferation is likely until something makes it impossible for anyone to make more nuclear weapons for a very long time.  

== Discussion == 

A growing number of leading figures have said that as long as the world maintains large nuclear arsenals, it is only a matter of time before there is a nuclear war.  Concerns like this have been expressed by two former US Secretaries of Defense ({{w|Robert McNamara}}&lt;ref&gt;{{cite Q|Q64736611}}&lt;!-- Robert McNamara and James G. Blight (2003) Wilson's ghost:  Reducing the risk of conflict, killing, and catastrophe in the 21st century --&gt;.&lt;/ref&gt; and {{w|William Perry}}, two former US Secretaries of State {{w|Henry Kissinger}} and {{w|George Schultz}}, former US Senator {{w|Sam Nunn}}&lt;ref&gt;{{cite Q|Q92101045}}&lt;!-- George P. Shultz, William J. Perry, and Sam Nunn, “The Threat of Nuclear War Is Still With Us”, WSJ 2019-04-10--&gt;.&lt;/ref&gt; and others with, for example, the {{w|Nuclear Threat Initiative}}. {{w|Daniel Ellsberg}} has said that a nuclear war will most likely generate a nuclear winter that lasts several years during which 98 percent of humanity will starve to death if they do not die of something else sooner.&lt;ref&gt;{{cite Q|Q64226035}}&lt;!-- Daniel Ellsberg and Amy Goodman and Juan González, “Daniel Ellsberg Reveals He was a Nuclear War Planner, Warns of Nuclear Winter &amp; Global Starvation”, 2017-12-06, Democracy Now!--&gt;.&lt;/ref&gt;

Banerjee and Duflo, two of the three who won the 2019 Nobel Memorial Prize in Economics, have noted that neither economic nor political stability are assured for any country, including the United States, China and India.  In particular, they predict that economic growth will almost certainly slow substantially in the latter two, leaving many poor people in desperate economic straits.&lt;ref&gt;{{cite Q|Q85764011}}&lt;!-- Abhijit V. Banerjee and Esther Duflo, Good Economics for Hard Times, 2019--&gt;. Various journalists and academic researchers have expressed concern about increases in ethic violence in various countries and whether electoral transitions of power will continue, even in the US.  See, e.g., {{cite Q|Q92101761}}&lt;!-- Brian Klaas, “Everyone knows the 2020 election will be divisive. But will it also be violent?”, Washington Post, 2019-09-05--&gt;.&lt;/ref&gt; Internal problems in the US, China, India or any other nuclear-weapon state could push political leaders to pursue increasingly risky foreign adventures, like Argentina did in 1982,&lt;ref&gt;{{w|Falklands War}}.&lt;/ref&gt; possibly leading to a war that could produce [[Time to nuclear Armageddon|nuclear Armageddon]].&lt;ref&gt;The risks of a nuclear war producing major global climate problems have been documented in a series of simulations published in refereed academic journals, each more detailed and more disconcerting than the previous.  All assume that many firestorms will be produced, because (a) the areas targeted will likely be much more susceptible to firestorms than the underground or isolated sites used to test nuclear weapons, and (b) many of the weapons used will have yields substantially greater than those employed in Hiroshima and Nagasaki.  For a discussion of that literature, see [[Time to extinction of civilization]] and [[Time to nuclear Armageddon]].&lt;/ref&gt;

The evidence compiled in the present work only seems to increase the urgency of limiting the threat of nuclear war and nuclear proliferation in particular.

In the 20 years following the first test of a nuclear weapon on 1945-07-16 by the US, four more nations acquired such weapons.  In the 50 years since the Non-Proliferation Treaty took effect in 1970, another four acquired them.&lt;ref&gt;This uses a commonly accepted list of existing nuclear-weapon states and when they each first tested a nuclear weapon.  The sources used for the data are in the help file for the “nuclearWeaponStates” dataset in the “Ecdat” package for R.  See {{cite Q|Q56452356}}&lt;!-- Ecdat: Data Sets for Econometrics --&gt;.&lt;/ref&gt; Our analysis of the available data considering only the dates of these first tests suggests that nuclear proliferation may have been slowing throughout this period.  However, that apparent trend was not statistically significant in the model we fit.  

Bayesian Model Averages (BMA) is known to generally produce better predictions than single model fits.  Accordingly, we've estimated confidence, prediction, and tolerance limits for the number of new nuclear-weapon states 74 years into the future based on two BMA models with mixtures of either a constant with a linear model or a constant with terms up to quartic in the time since the very first test of a nuclear weapon.   

We can expect that some non-nuclear nations and terrorist groups would eagerly pursue nuclear weapons if such seemed feasible unless some unprecedented change in international law provided them with effective nonviolent recourse to perceived threats.{{cn|cite Bacevich: Mutual Assured Destruction may not deter someone who thinks that Armageddon might be good.}}  

Moreover, these weapons will likely become more available with the passage of time unless (a) a nuclear war destroys everyone's ability to make more such weapons for a long time, or (b) there is a major change in the structure of international relations that has far more success than similar previous efforts in limiting the ability of new nations and non-state actors to acquire nuclear weapons.

== Monitoring nuclear proliferation ==

Organizations like the {{w|Wisconsin Project on Nuclear Arms Control}}, the {{w|Federation of American Scientists}}, the {{w|Stockholm International Peace Research Institute}}, and other similar organizations seem to have made substantive contributions to the apparent reduction in the rate of nuclear proliferation visible in most of the plots included in this article.  

In 2017 the Nuclear Verification Capabilities Independent Task Force of the {{w|Federation of American Scientists}} published seven recommendations for improving the process of nuclear monitoring and verification:&lt;ref&gt;{{cite Q
|Q97136193
}}&lt;/ref&gt;

# A network of 4-5 independent Centers of Nonproliferation Authentication
# The {{w|P5+1}} and Iran should publicize important implementation steps.
# Periodic public updates on monitoring &amp; U.S. support to the IAEA.
# The P5+1 and others should encourage Iranian openness.
# A trusted body of outside experts should be created to monitor the Iranian nuclear agreement.
# NGOs concerned with nonproliferation should aggressively protect both the information and the physical safety of its sources.
# Funders of nonproliferation NGOs should strengthen cyber security.

These seven recommendations seem likely to contribute to the trend towards a reduction in the rate of nuclear proliferation visible in many of the figures included in this article.  

However, the Federation of American Scientists was founded in 1946, and only one of the current nuclear-weapon states had such weapons before they were founded.  When Stockholm International Peace Research Institute was founded in 1966, there were five nuclear-weapon states.  When the Wisconsin Project on Nuclear Arms Control was founded in 1986, there were seven.  Two more nations have joined the list of nuclear-weapon states since this Wisconsin Project was founded.  Even if these seven recommendations are fully implemented, it seems unlikely that those actions by themselves will end nuclear proliferation.  We can hope that they will contribute slowing the rate of nuclear proliferation already implicitly considered in the model fit and forecasts discussed above.  Sadly the recent actions by the US and Russia in embarking on major &quot;modernization&quot; programs seem to be cause for concern.

== The Trump administration and nuclear weapons == 

Several actions of the [[w:Presidency of Donald Trump|Trump administration]] have raised concern about a new arms race, escalating bellecosity of the US in international relations and even possibly accelerating the threat of further nuclear proliferation.

* Terminated the 1987-88 {{w|Intermediate-Range Nuclear Forces Treaty}} alleging both Russian non-compliance and concerns about the continuing growth of China's missile forces.  This was announced 20 October 2018 and completed on 1 February 2019.  

* On May 21, 2020, President Trump announced that the United States would withdraw from the 2002 {{w|Treaty on Open Skies}}, alleging Russian violations.&lt;ref name='openSkies'&gt;{{Cite news
|last=Riechman
|first=Deb
|date=May 21, 2020
|title=US says it's pulling out of Open Skies surveillance treaty
|publisher=[[Associated Press]]
|url=https://apnews.com/773c5e6b7fb92f5e6d0e4b8fddf1665e
|url-status=live
|access-date=May 21, 2020
|archive-url=https://web.archive.org/web/20200521151337/https://apnews.com/773c5e6b7fb92f5e6d0e4b8fddf1665e
|archive-date=May 21, 2020
}}&lt;/ref&gt;

* The 1996 {{w|Comprehensive Nuclear-Test-Ban Treaty}} has been signed but not ratified by the US.  This makes it legal for the [[w:Presidency of Donald Trump|Trump administration]] to resume testing at any time.  They have reportedly been discussing conducting the first nuclear test since 1992.&lt;ref name='openSkies'/&gt;

* The Trump administration has expressed a desire to build smaller more &quot;usable&quot; nukes.  The use of such weapons by themselves seem less likely to produce a {{w|nuclear winter}} or autumn but could increase the chances of a full scale nuclear war using the larger weapons that would more likely produce [[Time to nuclear Armageddon|nuclear Armageddon]].{{cn}}  

* The Trump administration announced the sale of a nuclear reactor to Saudi Arabia.  Critics express concern that this would increase the risk that the Saudis may develop their own nuclear weapons.{{cn}} This in turn is particularly worrying for several reasons.  First, there is [[w:The 28 pages|substantial evidence that leading Saudis including members of the Saudi royal family and employees of the Saudi embassy and consulates in the US]] actively supported the preparations for the [[w:September 11 attacks|September 11 attacks]] of 2001.  Second, [[Winning the War on Terror|the vast majority of Islamic terrorists]] belong to the Wahabbi / Salafist branch of Islam, which is by far the most violent branch of Islam.  Third, the Saudi Government has continued to support {{w|al Qaeda}} at least as recently as 2019.&lt;ref name=SaudiQaeda/&gt;

== Conclusions ==

It seems likely that nuclear proliferation will continue until an international movement has far more success than similar previous efforts in ending it.  The seven recommendations of the Wisconsin Project on Nuclear Arms Control mentioned above may or may not slow nuclear proliferation enough to prevent nuclear Armageddon destroying civilization, dramatically shorten the lives of nearly all humans on earth.  

Might it be possible to energize existing organizations concerned about nuclear proliferation to the point that they have unprecedented success in achieving nearly complete nuclear disarmament and in strengthening international law so the poor, weak and disfranchised have effective nonviolent means for pursuing a redress of grievances?

== Appendix.  Companion R Markdown vignettes == 

Statistical details that make [[w:Reproducibility|the research in article reproducible]] are provided in two R Markdown vignettes on &quot;Forecasting nuclear proliferation&quot; and &quot;GDPs of nuclear-weapon states&quot;:  

* [[Forecasting nuclear proliferation/Simulating nuclear proliferation]]
* [[Forecasting nuclear proliferation/GDPs of nuclear-weapon states]]

== See also == 

* [[Time to extinction of civilization]]
* [[Time to nuclear Armageddon]]

== References == 

* {{cite Q|Q62670082}}&lt;!-- Burnham and Anderson (1998) Model selection and mutimodel inference --&gt;. 

* {{cite Q|Q27500468}}&lt;!-- Toon et al. (2007) Atmospheric effects and societal consequences of regional scale nuclear conflicts and acts of individual nuclear terrorism --&gt;

== Notes ==
{{reflist}}

[[Category:Original research]]
[[Category:Research]]
[[Category:Political science]]
[[Category:Military]]
[[Category:Military Science]]
[[Category:Freedom and abundance]]
[[Category:Reliability]]
[[Category:Reliability engineering]]
[[Category:Survival analysis]]
[[Category:Nuclear warfare]]</text>
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  <page>
    <title>Social Victorians/People/Rothschild Family</title>
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      <format>text/x-wiki</format>
      <text bytes="37764" xml:space="preserve">== Members ==
*Mr. Alfred de Rothschild
*Miss Alice de Rothschild
*Mr. and Mrs. Leopold de Rothschild

=== Titles ===
*Some of the Barons de or von Rothschild have a European, especially an Austrian, French, or Italian title. These are the U.K. titles.
**Baron de Rothschild [U.K.], created 29 June 1885&lt;ref name=&quot;:1&quot;&gt;&quot;Nathan Mayer de Rothschild, 1st Baron Rothschild.&quot; {{Cite web|url=https://www.thepeerage.com/p7037.htm#i70366|title=Person Page|website=www.thepeerage.com|access-date=2020-10-26}}&lt;/ref&gt;
***Nathan Mayer de Rothschild, 1st Baron Rothschild (29 June 1885 – 31 March 1915), the U.K. title
**1st Baronet Rothschild [U.K.], created 12 January 1847&lt;ref name=&quot;:2&quot;&gt;&quot;Sir Anthony Nathan Rothschild, 1st Bt.&quot; {{Cite web|url=https://www.thepeerage.com/p11819.htm#i118184|title=Person Page|website=www.thepeerage.com|access-date=2020-10-27}}&lt;/ref&gt;
***Sir Anthony Nathan Rothschild, 1st Bt. (12 January 1847 – 4 January 1876)
**1st Baron Battersea of Battersea, co. London and of Overstrand, Norfolk [U.K.], created 5 September 1892&lt;ref&gt;&quot;Cyril Flower, 1st and last Baron Battersea of Battersea.&quot; {{Cite web|url=https://www.thepeerage.com/p11810.htm#i118100|title=Person Page|website=www.thepeerage.com|access-date=2020-10-27}}&lt;/ref&gt;
***Cyril Flower, 1st and last Baron Battersea of Battersea (5 September 1892 – 27 November 1907)

== Acquaintances, Friends and Enemies ==
=== Friends ===

==== Ferdinand de Rothschild ====
*[[Social Victorians/People/Chamberlain | Joseph Chamberlain]]
*[[Social Victorians/People/Balfour | Arthur Balfour]]
*[[Social Victorians/People/Churchill | Randolph Churchill]]

== Organizations ==

=== Alfred Charles de Rothschild ===
* Bank of England, director&lt;ref&gt;&quot;Alfred Charles de Rothschild.&quot; {{Cite web|url=https://www.thepeerage.com/p19553.htm#i195527|title=Person Page|website=www.thepeerage.com|access-date=2020-10-26}}&lt;/ref&gt;

== Timeline ==
'''1840 March 30''', Sir Anthony Nathan Rothschild and Louisa Montefiore married.&lt;ref name=&quot;:5&quot;&gt;&quot;Louisa Montefiore.&quot; {{Cite web|url=https://www.thepeerage.com/p19548.htm#i195480|title=Person Page|website=www.thepeerage.com|access-date=2020-10-27}}&lt;/ref&gt;

'''1867 April 17''', Nathan Mayer de Rothschild and Emma Louisa Rothschild married.&lt;ref name=&quot;:4&quot;&gt;&quot;Emma Louise Rothschild.&quot; {{Cite web|url=https://www.thepeerage.com/p7107.htm#i71069|title=Person Page|website=www.thepeerage.com|access-date=2020-10-26}}&lt;/ref&gt;

'''1877 November 22''', Constance de Rothschild and Cyril Flower married.&lt;ref name=&quot;:6&quot;&gt;&quot;Constance de Rothschild.&quot; {{Cite web|url=https://www.thepeerage.com/p11819.htm#i118183|title=Person Page|website=www.thepeerage.com|access-date=2020-10-27}}&lt;/ref&gt;

'''1881 January 19''', Leopold de Rothschild and Marie Perugia married.&lt;ref name=&quot;:0&quot;&gt;&quot;Marie Perugia.&quot; {{Cite web|url=https://www.thepeerage.com/p19553.htm#i195530|title=Person Page|website=www.thepeerage.com|access-date=2020-10-26}}&lt;/ref&gt;

'''1885''', Nathaniel de Rothschild &quot;was sworn into the House of Lords on a copy of the Torah with his head covered.&quot;&lt;ref name=&quot;:14&quot;&gt;Simmons, Michael W. ''The Rothschilds: The Dynasty and the Legacy''.  Make Profits Easy, LLC, 17 January 2017.&lt;/ref&gt;{{rp|118 (of 194)}}

'''1897, sometime during the Jubilee ceremonies, probably June''', Nathaniel Rothschild played an official role:&lt;blockquote&gt;When Queen Victoria celebrated her Jubilee in 1897, the highest ranking Catholic cardinal in England gave an address in her honor on behalf of English Catholics; the corresponding address from English Jews might have been expected to come from England's chief Rabbi, but instead it was given by Lord Rothschild.&lt;ref name=&quot;:14&quot; /&gt;{{rp|143–144 (of 1940)}}&lt;/blockquote&gt;

'''1897 July 2, Friday''', a number of members of the extended Rothschild family attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, including Lady (Emma Louise von Rothschild) and Lord Rothschild (Nathan Mayer de Rothschild), Baron F. de Rothschild, Mr. and Mrs. L. Rothschild (possibly Leopold and Marie Perugia Rothschild), Baron Ferdinand de Rothschild, Alfred Rothschild (?), Cyril Flower, Lord Battersea, and Constance de Rothschild Flower, Lady Battersea as well as Mr. Louis Flower. Louisa, Lady de Rothschild also attended.

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], at least two women who might be called Lady Rothschild attended, I think. Because of the photographs in the album in the National Portrait Gallery, however, they can be distinguished. Louisa, Lady Rothschild was married in 1840, so she was born probably somewhere around 1920; her husband, Sir Anthony Nathan Rothschild, was born in 1810 and died in 1876; so she was possibly around 75 at the ball. Emma, Lady Rothschild and Nathan Mayer de Rothschild were married in 1867, a generation later; she was born in 1844, so she was 53 or so at the time of the ball; her portrait is not in the album. Emma, Lady Rothschild and Nathan, Lord Rothschild were in the first seating for supper, the highest status of the Rothschilds at the ball, in this social network in any case.

# Cyril Flower, Lord Battersea is #110
# Constance de Rothschild Flower, Lady Battersea, is #328
# [[Social Victorians/People/Rothschild Family#Emma, Lady Rothschild and Nathan Mayer, Lord Rothschild|Emma, Lady Rothschild]] is #112 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who attended]]
# Nathan Mayer de Rothschild, Lord Rothschild is at 216
# Louisa, Lady de Rothschild is at 674
# Baron Ferdinand de Rothschild is at 330
# Mr. L. Rothschild, possibly Leopold de Rothschild, is at 527
# Mrs. L. Rothschild, possibly Marie Perugia Rothschild, is at 528
# Alfred Rothschild is at 605
#Evelyn de Rothschild is at 669
#Anthony de Rothschild is at 670
[[File:Cyril-Flower-1st-Baron-Battersea-as-Lord-Hunsdon-in-the-Elizabethan-Procession.jpg|thumb|alt=Black-and-white photograph of a standing man richly dressed in an historical costume with a sword and garter|Cyril Flower, 1st Baron Battersea as Lord Hunsdon in the Elizabethan Procession. ©National Portrait Gallery, London.]]
=== Cyril Flower, Lord Battersea and Constance de Rothschild Flower, Lady Battersea ===
Cyril Flower, Lord Battersea was dressed
*&quot;in cerise and silver brocade copied from an old Jacquemin.&quot;&lt;ref name=&quot;:8&quot;&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and https://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 7a}}
*as Lord Hunsdon, a &quot;Gentleman of the Court of Queen Elizabeth, in cerise and silver brocade, copied from an old picture by Jacquemin.&quot;&lt;ref name=&quot;:7&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|p. 8, Col. 1c}}
*as &quot;(gentleman of the Court of Queen Elizabeth), in cerise and silver brocade; from an old picture by Jacquemin.&quot;&lt;ref name=&quot;:12&quot; /&gt;{{rp|36, Col. 3b}}
Elliott &amp; Fry's portrait of &quot;Cyril Flower, 1st Baron Battersea as Lord Hunsdon in the Elizabethan Procession&quot; in costume is photogravure #74 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:10&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;Lord Battersea as Lord Hunsdon in the Elizabethan procession,&quot; with a Long S in ''procession''.&lt;ref&gt;&quot;Cyril Flower, 1st Baron Battersea as Lord Hunsdon in the Elizabethan Procession.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158430/Cyril-Flower-1st-Baron-Battersea-as-Lord-Hunsdon-in-the-Elizabethan-Procession.&lt;/ref&gt;

Constance de Rothschild Flower, Lady Battersea, was dressed as Lady Hunsdon in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#Queen_Elizabeth_Court |Queen Elizabeth procession]].
*Lady Battersea &quot;was perfectly dressed as a lady of the Elizabethan period, in a cream velvet train and bodice, with stomacher and front of red velvet, the latter studded with exquisite pearls. This was trimmed with old silver galon and a Medici collar of lace embroidered in pearls.&quot;&lt;ref name=&quot;:8&quot; /&gt;{{rp|p. 6, Col. 1b}}
*&quot;Lady Battersea was dressed as a lady of the Elizabethan period, in a cream velvet train and bodice, with stomacher and front of red velvet, the latter studded with exquisite pearls. This was trimmed with old silver galon and a Medici collar of lace embroidered in pearls.&quot;&lt;ref name=&quot;:13&quot;&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 4b}}
*&quot;Lady Battersea was, perhaps, the best dressed of the ladies of the Elizabethan period.&quot;&lt;ref&gt;“The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.&lt;/ref&gt;{{rp|p. 5, Col. 1}}
*&quot;Lady Battersea was perfect [sic?] dressed as a lady of the Elizabethan period in a cream velvet train and bodice, with stomacher and front of red velvet, the latter studded with exquisite pearls. This was trimmed with old silver gallon, and a Medici collar of lace embroidered in pearls.&quot;&lt;ref name=&quot;:9&quot;&gt;&quot;The Duchess of Devonshire's Fancy Dress Ball. Special Telegram.&quot; ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9c [of 9]–6, Col. 1a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.&lt;/ref&gt;{{rp|p. 6, Col. 1b}}
*&quot;Lady Battersea wore a beautiful Elizabethan costume, a cream velvet train, the bodice having a stomacher of red velvet, pearls, and a Medici collar, and was in the Queen Elizabeth procession.&quot;&lt;ref&gt;Holt, Ardern. “Dress and Fashion. To Correspondents.” The ''Queen'', The Lady’s Newspaper 31 July 1897, Saturday: 52 [of 84], Col. 1c [3 of 3 cols.]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18970731/321/0052.&lt;/ref&gt;

=== Mr. Lewis Flower ===
Cyril Flower's brother Lewis Peter Flower seems a likely candidate for the man whom the ''Gentlewoman'' calls Mr. Louis Flower. Mr. Flower (at 506) was dressed as a &quot;French Commissary General, First Empire&quot; in a &quot;regimental coat, slightly open at neck, showing stock, dark blue cloth, collar and cuffs scarlet, embroidered in silver; riding breeches, buff cloth; large cocked hat of the First Empire, bound with black braiding; tricolour badge and silver cord on left side.&quot;&lt;ref name=&quot;:12&quot; /&gt;{{rp|p. 42, Col. 1a}}

=== Emma, Lady Rothschild and Nathan Mayer, Lord Rothschild ===
Emma Louise von Rothschild, Lady Rothschild, dressed as Anne of Cleves, sat at Table 11 in the first seating for supper, escorted in by the Earl of Suffolk.
*She was dressed as &quot;Anne of Cleves. Bodice in black velvet, black satin skirt and facings, petticoat and undersleeves in old blue and gold brocade, embroidered in gold. All the jewels were real old ones of the period.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|p. 8, Col. 1c}}
*&quot;Lady Rothschild’s costume, as Anne of Cleves, looked beautiful with its petticoat and undersleeves of old blue and gold brocade, embroidered with gold, and its black satin skirt and bodice of black velvet. All Lady Rothschild’s jewels were authentic antique gems of the period.&quot;&lt;ref&gt;“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.&lt;/ref&gt;
*&quot;Lady Rothschild went as Anne of Cleves in black velvet and satin, relieved with antique brocade and gold embroidery. Her jewellery was all antique.&quot;&lt;ref&gt;“Girls’ Gossip.” ''Truth'' 8 July 1897, Thursday: 41 [of 70], Col. 1b – 42, Col. 2c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18970708/089/0041.&lt;/ref&gt;{{rp|42, Col. 1c}}
*&quot;Lady Rothschild, as Anne of Cleves, wore one of the most splendid dresses in the room; over a petticoat of old gold and blue hung an overmantle of satin literally encrusted with gold and precious stones, which were taken from her own museum, and which were actually the jewels of the period she represented.&quot;&lt;ref name=&quot;:12&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 32, Col. 3b}}
*&quot;L&lt;small&gt;ADY&lt;/small&gt; R&lt;small&gt;OTHSCHILD&lt;/small&gt;, as Anne of Cleves, wore a bodice in black velvet, black satin skirt and facings; petticoat and undersleeves in old blue and gold brocade, embroidered with gold; all the jewels being real old ones of the period.&quot;&lt;ref name=&quot;:17&quot;&gt;“Additional Costumes Worn at the Duchess of Devonshire’s Fancy Ball.” The ''Queen, The Lady’s Newspaper''17 July 1897, Saturday: 63 [of 97 BNA; p. 138 on the print page], Col. 2a–3a [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970717/283/0064.&lt;/ref&gt;{{rp|Col. 3a}}

The ''Queen'' published a line drawing signed by Rook of Lady Rothschild in costume as Anne of Cleves; the caption says [[Social Victorians/People/Dressmakers and Costumiers#Nathan|Messrs. Nathan]] made her dress.&lt;ref name=&quot;:17&quot; /&gt;{{rp|Col. 2b–c}}

Nathan Mayer de Rothschild, Lord Rothschild (#216) sat at Table 12 in the first seating for supper and was dressed as a Swiss Burgher in the Queen Elizabeth procession.
[[File:Hans Holbein the Younger (after) - Elizabeth Vaux (Royal Collection).jpg|thumb|alt=|Elizabeth Vaux, after Hans Holbein the Younger, painted c. 1600 – c. 1630]]

=== Louisa, Lady de Rothschild ===
[[File:Louisa-ne-Montefiore-Lady-de-Rothschild-as-Lady-Vaux-after-Holbein.jpg|thumb|left|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume|Louisa, Lady de Rothschild in costume as Lady Vaux (after Holbein). ©National Portrait Gallery, London.]]Louisa Montefiore, Lady de Rothschild (at 674) was the widow of Sir Anthony Nathan Rothschild, 1st Bt., who died in 1876.

Henry Van der Weyde's portrait of &quot;Louisa (née Montefiore), Lady de Rothschild as Lady Vaux (after Holbein)&quot; in costume is photogravure #182 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:10&quot; /&gt; The printing on the portrait says, &quot;Lady Rothschild as Lady Vaux (after Holbein).&quot;&lt;ref&gt;&quot;Lady Rothschild as Lady Vaux.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158545/Louisa-ne-Montefiore-Lady-de-Rothschild-as-Lady-Vaux-after-Holbein.&lt;/ref&gt;

==== The Historical Lady Vaux ====
Elizabeth Cheyne (or Cheney), Lady Vaux (1505–1556) was married to Thomas Vaux, 2nd Baron Vaux of Harrowden (1509 – 1556), an Elizabethan poet.&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|date=2021-09-25|title=Thomas Vaux, 2nd Baron Vaux of Harrowden|url=https://en.wikipedia.org/w/index.php?title=Thomas_Vaux,_2nd_Baron_Vaux_of_Harrowden&amp;oldid=1046392388|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Thomas_Vaux,_2nd_Baron_Vaux_of_Harrowden.&lt;/ref&gt; She was first cousin of Catherine Parr, Henry VIII's sixth wife.&lt;ref name=&quot;:15&quot; /&gt; This portrait of her is in the Royal Collection at Hampton Court, which is where Louisa, Lady de Rothschild would likely have seen it.

==== Newspaper Reports ====
Like others of the Rothschilds, Louisa, Lady Rothschild attended the ball in a costume that was much admired:

* &quot;[S]uch dresses as those of Lady Rothschild, after Holbein's Lady Vaux, of Messrs. Ferdinand and Alfred Rothschild, as an Austrian and French noble of the 16th century, were of extraordinary truth and beauty.&quot;&lt;ref name=&quot;:11&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;
* &quot;Lady Rothschild was beautifully attired as a Lady of the Tudor period, after a picture by Holbein.&quot;&lt;ref name=&quot;:8&quot; /&gt;{{rp|p. 6, Col. 1b}} &lt;ref name=&quot;:13&quot; /&gt;{{rp|p. 3, Col. 4b}}
*&quot;Lady Rothschild was beautifully attired as a lady of the Tudor period, after a picture by Holbein.&quot;&lt;ref name=&quot;:9&quot; /&gt;{{rp|p. 6, Col. 1b}}

=== Baron Ferdinand de Rothschild ===
[[File:Ferdinand-James-Anselm-de-Rothschild-Baron-de-Rothschild-as-Casimir-Count-Palatine.jpg|thumb|left|alt=Black-and-white photograph of a standing man richly dressed in an historical costume|Baron Ferdinand de Rothschild in costume as Casimir Count Palatine. ©National Portrait Gallery, London.]]
[[File:Johann Casimir aus Thesaurus Pictuarum.jpg|thumb|alt=old portrait of a standing man with gloves, sword, cloak and hat|Johann Casimir]]
Baron F. de Rothschild or Baron Ferdinand de Rothschild (at 330) was the widower of Evelina de Rothschild, who had died in 1866. He attended the ball dressed as Casimir Count Patatine of Bavaria in the Queen Elizabeth procession. Like others of the Rothschilds, his costume was notable.
*&quot;Baron F. de Rothschild appeared as Casimir, Count Palatine of Bavaria.&quot;&lt;ref&gt;“Fancy Dress Ball at Devonshire House. A Brilliant Spectacle.” ''Derbyshire Advertiser and Journal'' 10 July 1897, Saturday: 6 [of 8], Cols. 5a–6a. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0001084/18970710/136/0006.&lt;/ref&gt;
*Baron F. de Rothschild was dressed &quot;as an Austrian noble of the 16th century.&quot;&lt;ref name=&quot;:11&quot; /&gt;
*&quot;[S]uch dresses as those of ... Messrs. Ferdinand and Alfred Rothschild, as an Austrian and French noble of the 16th century, were of extraordinary truth and beauty.&quot;&lt;ref name=&quot;:11&quot; /&gt;
Lafayette's portrait of &quot;Ferdinand James Anselm de Rothschild, Baron de Rothschild as Casimir Count Palatine&quot; in costume is photogravure #218 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:10&quot; /&gt; The printing on the portrait says, &quot;Baron Ferdinand de Rothschild as Casimir Count Palatine.&quot;&lt;ref&gt;&quot;Baron Ferdinand de Rothschild as Casimir Count Palatine.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158581/Ferdinand-James-Anselm-de-Rothschild-Baron-de-Rothschild-as-Casimir-Count-Palatine.&lt;/ref&gt;

The portrait (right) of Ioannes Casimirvs Palatinus Rheini Dux Bavaria is from Thesaurus Picturarum, no earlier than 1568.
=== Leopold de Rothschild and Marie Perugia Rothschild ===
[[File:Marie-Rothschild-ne-Perugia-as-Zobeida.jpg|thumb|left|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume|Marie Rothschild (née Perugia) in costume as Zobeida. ©National Portrait Gallery, London.]][[File:Leopold-de-Rothschild-as-Duc-de-Sully.jpg|thumb|alt=Black-and-white photograph of a standing man richly dressed in an historical costume with a sword, cape and plumed hat|Leopold de Rothschild in costume as Duc de Sully. ©National Portrait Gallery, London.]]
Mrs. Leopold (Marie Perugia) Rothschild (#528): Alice Hughes's portrait of &quot;Marie Rothschild (née Perugia) as Zobeida&quot; in costume is photogravure #114 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:10&quot; /&gt; The printing on the portrait says, &quot;Mr. Leopold de Rothschild as Zobeida.&quot;&lt;ref&gt;&quot;Marie Rothschild (née Perugia).&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158476/Marie-Rothschild-ne-Perugia-as-Zobeida.&lt;/ref&gt;

Zubaidah bint Ja`far ibn Mansur ( – 831, C.E.) was an Abbasid princess and queen who managed her properties and wealth independent of her husband, Harun al-Rashid.&lt;ref&gt;{{Cite journal|date=2021-08-21|title=Zubaidah bint Ja'far|url=https://en.wikipedia.org/w/index.php?title=Zubaidah_bint_Ja%27far&amp;oldid=1039860950|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Zubaidah_bint_Ja%27far.&lt;/ref&gt;

Mr. Leopold de Rothschild (#527): John Thomson's portrait of &quot;Leopold de Rothschild as Duc de Sully&quot; in costume is photogravure #115 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:10&quot; /&gt; The printing on the portrait says, &quot;Mr. Leopold de Rothschild as Duc de Sully.&quot;&lt;ref&gt;&quot;Leopold de Rothschild.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158477/Leopold-de-Rothschild-as-Duc-de-Sully.&lt;/ref&gt;

Maximilien de Béthune, 1st Duke of Sully (1560–1641) was an advisor to French King Henry IV and noted for being a good businessman.&lt;ref&gt;{{Cite journal|date=2021-07-04|title=Maximilien de Béthune, Duke of Sully|url=https://en.wikipedia.org/w/index.php?title=Maximilien_de_B%C3%A9thune,_Duke_of_Sully&amp;oldid=1031909919|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Maximilien_de_Béthune,_Duke_of_Sully.&lt;/ref&gt; Leopold Rothschild is wearing insignias for some orders in this portrait.

[[File:Evelyn-Achille-de-Rothschild-as-a-page-to-the-Doges-Wife.jpg|thumb|left|alt=Black-and-white photograph of a standing boy richly dressed in an historical costume with a cape and ruff around his neck|Evelyn de Rothschild, as a page to the Doge's Wife. ©National Portrait Gallery, London.]]
Two children of Leopold and Marie de Rothschild also attended, as pages of the &quot;wife of the Doge,&quot; Evelyn Achille de Rothschild (#669) and Anthony Gustav de Rothschild (#670). Their portraits appear in the Album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:10&quot; /&gt; One person is described as Dogaressa in the Album: [[Social Victorians/People/Arthur Sassoon | Louise (Mrs. Arthur) Sassoon]] (at 202); two men are called doges: [[Social Victorians/People/Wilbraham Egerton of Tatton | Lord Wilbraham Egerton of Tatton]] (at 591) and Edward Bootle-Wilbraham, the [[Social Victorians/People/Lathom | Earl of Lathom]] (at 125).

[[File:Anthony-Gustav-de-Rothschild-as-a-page-to-the-Doges-Wife.jpg|thumb|alt=Black-and-white photograph of a boy sitting on a window seat, richly dressed in an historical costume as a page, with a large hat next to him, a cape and a ruff around his neck|Anthony de Rothschild, as a page to the Doge's Wife. ©National Portrait Gallery, London.]]
=== Evelyn and Anthony de Rothschild ===

Evelyn de Rothschild (at 669) and Anthony de Rothschild (at 670) were pages, attending [[Social Victorians/People/Arthur Sassoon|Louise (Mrs. Arthur) Sassoon]]. They were too young, probably, to have been invited in their own right: Evelyn was 11 and Anthony was 10 years old.

Alice Hughes's portrait of &quot;Evelyn Achille de Rothschild as a page to the Doge's Wife&quot; in costume is photogravure #165 in the album. The printing on the portrait says, &quot;Master Evelyn de Rothschild as a page to the Doge's Wife.&quot;&lt;ref&gt;&quot;Master Evelyn de Rothschild.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158528/Evelyn-Achille-de-Rothschild-as-a-page-to-the-Doges-Wife.&lt;/ref&gt;

Alice Hughes's portrait of &quot;Anthony Gustav de Rothschild as a page to the Doge's Wife&quot; in costume is photogravure #166 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:10&quot; /&gt; The printing on the portrait says, &quot;Master Anthony de Rothschild as a page to the Doge's Wife.&quot;&lt;ref&gt;&quot;Master Anthony de Rothschild.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158529/Anthony-Gustav-de-Rothschild-as-a-page-to-the-Doges-Wife.&lt;/ref&gt;

[[File:Alfred-Charles-de-Rothschild-as-King-Henry-III.jpg|thumb|left|alt=Black-and-white photograph of a standing man richly dressed in an historical costume|Alfred de Rothschild in costume as King Henry III. ©National Portrait Gallery, London.]]
=== Alfred Rothschild ===
[[File:Louis9+Henri3+StDenis.jpg|thumb|alt=Old panel showing King Henry the 3rd of England in two scenes|Henry III visiting Louis IX of France (left) and visiting St. Denis (right)]]
Alfred Rothschild (at 605) was present as well in a costume described variously, from King Henry III of England to a 16th-century French and Austrian noble. 

* Alfred Rothschild was dressed &quot;as [a] French noble of the 16th century.&lt;ref name=&quot;:11&quot; /&gt;
* &quot;[S]uch dresses as those of ... Messrs. Ferdinand and Alfred Rothschild, as an Austrian and French noble of the 16th century, were of extraordinary truth and beauty.&quot;&lt;ref name=&quot;:11&quot; /&gt;

John Thomson's portrait of &quot;Alfred Charles de Rothschild as King Henry III&quot; in costume is photogravure #224 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:10&quot; /&gt; The printing on the portrait says, &quot;Mr. Alfred Charles de Rothschild as Henry III.&quot;&lt;ref&gt;&quot;Alfred Charles de Rothschild as Henry III.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158587/Alfred-Charles-de-Rothschild-as-King-Henry-III.&lt;/ref&gt;

No available portraits of King Henry III of England (1207–1272)&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2021-12-27|title=Henry III of England|url=https://en.wikipedia.org/w/index.php?title=Henry_III_of_England&amp;oldid=1062218783|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Henry_III_of_England.&lt;/ref&gt; show him dressed in anything similar to what Alfred Rothschild is wearing in his portrait. The panel (right) with two scenes of Henry III is from an illuminated manuscript now held by the British Museum, which acquired it in 1757, when King George II presented it to them.&lt;ref&gt;&quot;Detailed Record.&quot; {{Cite web|url=https://www.bl.uk/catalogues/illuminatedmanuscripts/ILLUMIN.ASP?Size=mid&amp;IllID=43001|title=Image of an item from the British Library Catalogue of Illuminated Manuscripts|last=Wight|first=C.|website=www.bl.uk|access-date=2021-12-28}} https://www.bl.uk/catalogues/illuminatedmanuscripts/ILLUMIN.ASP?Size=mid&amp;IllID=43001.&lt;/ref&gt; The costume looks more like a late-Victorian interpretation of 16th-century clothing than it does of 13th-century clothing.

Early in the reign of King Henry III of England, policies protected Jews in England, mostly for the financial benefit of the English, but by 1239 he had begun to put anti-Jewish policies into place.&lt;ref name=&quot;:16&quot; /&gt;

== Snapshots ==

=== Leopold Rothschild ===
Brother of Nathaniel and Alfred, of the English branch. A general sketch of Leopold:&lt;blockquote&gt;Outside the office, Leopold was primarily occupied with horses, while Alfred was a connoisseur of the arts. Though Leo de Rothschild was one of the first people ever to own an automobile, and was also a famous breeder and racer of horses, he was best known for his kindness, which was characterized not just by the expenditure of large sums of money, but by tremendous thoughtfulness. One Rothschild employee wrote that Leo &quot;devoted himself to the welfare of the clerks, not so much as a duty or in a spirit of noblesse oblige as because it was in his character to do so ... [sic] One man suffering from a chest complaint ws sent by Leopold to Australia for six months; another, distraught by the death of his wife, was given a sea trip around the world.&lt;ref name=&quot;:14&quot; /&gt;{{rp|p. 121 [of 194]}}&lt;/blockquote&gt;

=== Alfred Rothschild ===
Brother of Nathaniel and Leopold, of the English branch. Alfred in general:&lt;blockquote&gt;Alfred de Rothschild, by comparison, was more of a typical late Victorian aesthete [than Leopold]. Unusually for a Rothschild, he was slender, slight, blonde haired and blue eyed. He employed himself principally in the pursuit of luxury — music, fine clothes, antique furniture, paintings, etc.  Yet his most enduring legacy was the French chateau he built in Buckinghamshire, called Halton House. Despite the fact that tAlfred was one of the most highly regarded art experts in England, Halton House was famous for its ugliness. &quot;An exaggerated nightmare of gorgeousness / and senseless and ill-applied magnificent,&quot; wrote one guest. &quot;I have seldom seen anything more terribly vulgar,&quot; wrote another. &quot;Outside it is a combination of a French chateau and a gambling house. Inside it is badly planned, gaudily decorated ... O, but the hideousness of everything, the showiness! The sense of lavish wealth thrust up your nose! The coarse moldings, the heavy gildings always in the wrong place, the color of the silk hangings! Eye hath not seen nor pen can write the ghastly coarseness of the sight!&quot;&lt;ref name=&quot;:14&quot; /&gt;{{rp|pp. 123–124 [of 194]}}&lt;/blockquote&gt;Daisy, Lady Warwick wrote about Alfred Rothschild:&lt;blockquote&gt;In the famous white drawing-room at Seamore Place I have heard the greatest artistes in the world, who were paid royal fees to entertain a handful of his friends. Unfortunately, he could not share in the hospitality that he lavished upon those he esteemed, for he suffered from some obscure form of dyspepsia which no doctor could cure. Many a time I have seen him sit at the head of the table, exercising all the graces of a host, while he himself took neither food nor wine.

He used to ride every morning in the park, followed by his brougham. Park-keepers soon learnt how generous the millionaire was; they used to put stones on the road by which he would enter, then, when he came in sight, they would hasten to removed [sic] them — a courtesy which was invariably rewarded. He was shrewd enough to know just how the stones got there, but this childish / device amused him, so he pretended ignorance.&lt;ref name=&quot;:14&quot; /&gt;{{rp|pp. 124–125 [of 194]}}&lt;/blockquote&gt;

== Demographics ==
Anselm Salomon von Rothschild
*Nationality: Austrian
*Branch of the family: Vienna branch

=== Residences ===
* Anselm Salomon von Rothschild and then his unmarried younger sister Alice Charlotte Rothschild (1847–1922): Waddesdon Manor, Buckinghamshire&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|date=2020-08-16|title=Anselm Salomon von Rothschild|url=https://en.wikipedia.org/w/index.php?title=Anselm_Salomon_von_Rothschild&amp;oldid=973334060|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Cyril and Constance Flower, Lord and Lady Battersea: Aston Clinton, Buckinghamshire, England

== Family ==
*Anselm Salomon von Rothschild, baron (29 January 1803 – 27 July 1874)&lt;ref name=&quot;:3&quot; /&gt;
*Charlotte Nathan Rothschild (1807–1859)
#Mayer Anselm Leon (1827–1828)
#&quot;Julie&quot; Caroline Julie Anselm (1830–1907), married Adolph Carl von Rothschild (1823–1900), son of Carl Mayer von Rothschild at Naples
#Mathilde Hannah von Rothschild (1832–1924), married ''Freiherr'' Wilhelm Carl von Rothschild (1828–1901)
#Sarah Luisa (1834–1924), married Baron Raimondo Franchetti (1829–1905)
#Nathaniel Anselm (1836–1905)
#Ferdinand James (1839–1898)
#Albert Salomon (1844–1911)
#Alice Charlotte (1847–1922)


*Baron Lionel Nathan de Rothschild (22 November 1807 – 3 June 1879)&lt;ref&gt;&quot;Baron Lionel Nathan de Rothschild.&quot; {{Cite web|url=https://www.thepeerage.com/p19553.htm#i195523|title=Person Page|website=www.thepeerage.com|access-date=2020-10-26}}&lt;/ref&gt;
*Charlotte de Rothschild (1819 – 13 March 1884)&lt;ref&gt;&quot;Charlotte de Rothschild.&quot; {{Cite web|url=https://www.thepeerage.com/p13776.htm#i137756|title=Person Page|website=www.thepeerage.com|access-date=2020-10-26}}&lt;/ref&gt;
#'''Evelina de Rothschild''' (1839 – 4 December 1866)
#Leonora de Rothschild ( – 6 January 1911)
#Nathan Mayer de Rothschild, 1st Baron Rothschild (8 November 1840 – 31 March 1915)
#'''Alfred Charles de Rothschild''' (20 July 1842 – 31 January 1918)
#Leopold de Rothschild (22 November 1845 – 29 May 1917)


*Evelina de Rothschild (1839 – 4 December 1866)
*Ferdy (Ferdinand James Anselm), Freiherr von Rothschild (17 December 1839 – 17 December 1898)&lt;ref&gt;{{Cite journal|date=2020-08-16|title=Ferdinand de Rothschild|url=https://en.wikipedia.org/w/index.php?title=Ferdinand_de_Rothschild&amp;oldid=973342192|journal=Wikipedia|language=en}}&lt;/ref&gt;


*Natty (Nathan Mayer) de Rothschild, 1st Baron Rothschild (8 November 1840 – 31 March 1915)&lt;ref name=&quot;:1&quot; /&gt;
*Emma Louise von Rothschild (1844 – January 1935)&lt;ref name=&quot;:4&quot; /&gt;
#Walter (Lionel Walter) Rothschild, 2nd Baron Rothschild (8 February 1868 – 27 August 1937)
#Charlotte Louisa Adela Evelina Rothschild (3 April 1873 – 9 May 1947)
#Nathaniel Charles Rothschild (9 May 1877 – 12 October 1923)


*Leopold de Rothschild (22 November 1845 – 29 May 1917)
*Marie Perugia Rothschild (1862-1937)&lt;ref name=&quot;:0&quot; /&gt;
#Lionel Nathan de Rothschild (25 January 1882 – 28 January 1942)
#'''Evelyn Achille de Rothschild''' (6 January 1886 – )
#'''Anthony Gustav de Rothschild''' (26 June 1887 – 5 February 1961)


*Sir Anthony Nathan Rothschild, 1st Bt. (29 May 1810 – 4 January 1876)&lt;ref name=&quot;:2&quot; /&gt;
*Louisa Montefiore ( – 22 September 1910)&lt;ref name=&quot;:5&quot; /&gt;
#'''Constance de Rothschild''' (1843 – 22 November 1931)
#Annie Rothschild (1844 – 21 November 1926)


*Constance de Rothschild (1843 – 22 November 1931)&lt;ref name=&quot;:6&quot; /&gt;
*'''Cyril Flower, 1st and last Baron Battersea of Battersea''' (30 August 1843 – 27 November 1907)


* Walter (Lionel Walter) Rothschild, 2nd Baron Rothschild (8 February 1868 – 27 August 1937)&lt;ref&gt;&quot;Lionel Walter Rothschild, 2nd Baron Rothschild.&quot; {{Cite web|url=https://www.thepeerage.com/p7108.htm#i71071|title=Person Page|website=www.thepeerage.com|access-date=2020-10-26}}&lt;/ref&gt; (unmarried relationship)
* Marie Barbara Fredenson ()&lt;ref&gt;&quot;Marie Barbara Fredenson.&quot; {{Cite web|url=https://www.thepeerage.com/p21748.htm#i217473|title=Person Page|website=www.thepeerage.com|access-date=2020-10-27}}&lt;/ref&gt;
*# Olga Alice Muriel Rothschild ( – 1992)

=== Family of Cyril Flower ===

* Philip William Flower (1809 – 22 February 1872)&lt;ref&gt;&quot;Philip William Flower.&quot; {{Cite web|url=https://www.thepeerage.com/p11818.htm#i118180|title=Person Page|website=www.thepeerage.com|access-date=2021-06-05}} https://www.thepeerage.com/p11818.htm#i118180.&lt;/ref&gt;
* Mary Flower (1816–1857)&lt;ref&gt;&quot;Mary Flower.&quot; {{Cite web|url=https://www.thepeerage.com/p11819.htm#i118181|title=Person Page|website=www.thepeerage.com|access-date=2021-06-05}} https://www.thepeerage.com/p11819.htm#i118181.&lt;/ref&gt;
*# Hugh Philip Flower (1842–1862)
*# '''Cyril Flower, 1st and last Baron Battersea of Battersea''' (30 August 1843 – 27 November 1907)
*# Arthur Flower (1847 – 1 March 1911)
*# Clara Flower (1849–1871)
*# Horace Flower (1850–1885)
*# Augustus Flower (1851–1863)
*# Herbert Flower (1853 – 30 December 1881)
*# Alfred Flower (1854–1855)
*# '''Lewis Peter Flower''' (1856 – December 1902)

=== Relations ===

* Marie Perugia Rothschild's sister was [[Social Victorians/People/Arthur Sassoon|Louise Perugia Sassoon]]; Marie Rothschild's sons Evelyn Rothschild and Anthony Rothschild attended Louise Sassoon as her pages for the Duchess of Devonshire's 1897 fancy-dress ball.

== Memoirs, Autobiographies, Biographies, Personal Papers ==

=== Personal Papers ===

==== Lady Louise de Rothschild ====

# &quot;289. ''1837'' ROTHSCHILD, ''Lady'' Louisa de, philanthropist. Diary, 29 July 1837–2 Dec. 1907, with gaps, of family life in London, court life, music, travel, sport. BL Add MSS 47,949–62.&quot;&lt;ref&gt;&quot;Unpublished London Diaries.&quot; {{Cite web|url=https://www.british-history.ac.uk/london-record-soc/vol37/pp22-46|title=Checklist of Unpublished Diaries: nos 1-294 {{!}} British History Online|website=www.british-history.ac.uk|access-date=2024-04-25}} https://www.british-history.ac.uk/london-record-soc/vol37/pp22-46.&lt;/ref&gt; [Is BL British Library or Bodleian Library? If Bodleian, then it may be in the MSS Fisher collection.]

== Questions and Notes ==
#Baron Lionel Nathan de Rothschild and Sir Anthony Nathan Rothschild were brothers.
#Natty (Nathan Mayer) de Rothschild, 1st Baron Rothschild and [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]], met and became friends at Cambridge; neither graduated.&lt;ref&gt;{{Cite journal|date=2020-07-31|title=Nathan Rothschild, 1st Baron Rothschild|url=https://en.wikipedia.org/w/index.php?title=Nathan_Rothschild,_1st_Baron_Rothschild&amp;oldid=970472427|journal=Wikipedia|language=en}}&lt;/ref&gt;
#I'm guessing that Mr. and Mrs. L. Rothschild were Leopold and Marie Perugia Rothschild because Lionel Walter Rothschild, 2nd Baron Rothschild, Nathan and Emma's son, never married.
#The letterpress on the portrait in the Album in the NPG calls Louisa (née Montefiore), Lady de Rothschild Lady Rothschild.
#Mayer Amschel Rothschild (23 February 1744 – 19 September 1812), who founded the banking and family dynasty, stipulated that only male descendants could take part in the family business, which meant that female descendants had to marry cousins in order to stay part of it.&lt;ref name=&quot;:14&quot; /&gt; (50 [of 194]) He also stipulated Rothschild children were to marry 1st or 2nd cousins, though some — like Anselm Salomon von Rothschild's sister Betty, who married her uncle — married other near relatives.&lt;ref name=&quot;:3&quot; /&gt;
== Footnotes ==
{{reflist}}</text>
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  </page>
  <page>
    <title>Social Victorians/People/Paget Family</title>
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      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
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      <comment>/* Overview */</comment>
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      <text bytes="15471" xml:space="preserve">== Also Known As ==
*Family name: Paget
*Sir Arthur Fitzroy and Mrs. Minnie (Mary Stevens) Paget
**Mr. Fitzroy (nom de plume)
**General Rt. Hon. Sir Arthur Henry Fitzroy Paget
*Almeric Paget
*Mr. and Mrs. Cecil Paget
*Mr. and Mrs. George Ernest Paget
**and Miss Hylda Paget
*Mr. Gerald and Mrs. Lucy Paget

== Overview ==

* The subsequent son of a subsequent son, the 7th son of a landed general, Almeric Fitzroy was a civil servant. David Cannadine describes him as an example of &quot;landed-establishment life&quot; and one of the &quot;genteel mandarins&quot;:&lt;blockquote&gt;He was a great grandson of the third Duke of Grafton, and his mother was a daughter of Lord Feversham. He began his official life as an Inspector of Schools in the Education Department of the Privy Council. The appointment was arranged by family influence, and it gave Fitzroy time to hunt three days in every fortnight.... In 1884, [[Social Victorians/People/Carlingford|Lord Carlingford]] transferred him to the Privy Council Office itself; in 1895 the Duke of Devonshire (who had just become Lord President) made him his private secretary; and three years later, the combination of family influence and the Duke's patronage brought him the Clerkship of the Privy Council, which he held until his retirement in 1923. Throughout this period, he was on the closest terms with the leading politicians of the day, he moved easily in royal and patrician society, he was a well-known figure in the clubs of London, and he spent many a weekend at Chatsworth, Lissadell, Osterley, Longleat, and Euston.&lt;ref name=&quot;:8&quot;&gt;Cannadine, David. ''The Decline and Fall of the British Aristocracy''. New York: Yale University Press, 1990.&lt;/ref&gt;{{rp|242}}&lt;/blockquote&gt;

* Also, according to Cannadine, &quot;Almeric Fitzroy wrote books about his ancestors, and was a trustee of the Duke of Grafton's settlement.&quot;&lt;ref name=&quot;:8&quot; /&gt;{{rp|242}}

== Acquaintances, Friends and Enemies ==

== Timeline ==
'''1877 January 2''', Gerald Cecil Stewart Paget and Lucy Annie Emily Gardner married.&lt;ref name=&quot;:6&quot;&gt;&quot;Lucy Annie Emily Gardner.&quot; {{Cite web|url=https://www.thepeerage.com/p4699.htm#i46982|title=Person Page|website=www.thepeerage.com|access-date=2021-12-01}} https://www.thepeerage.com/p4699.htm#i46982.&lt;/ref&gt;

'''1878 July 27''', Mary Stevens and Arthur Henry Fitzroy Paget married.&lt;ref name=&quot;:3&quot;&gt;&quot;Mary Stevens.&quot; {{Cite web|url=https://www.thepeerage.com/p3392.htm#i33914|title=Person Page|website=www.thepeerage.com|access-date=2020-10-18}}&lt;/ref&gt;

'''1889 June 17''', Alexandra Harriet Paget and [[Social Victorians/People/Colebrooke|Edward Arthur Colebrooke]] married.&lt;ref&gt;&quot;Alexandra Harriet Paget.&quot; {{Cite web|url=https://www.thepeerage.com/p880.htm#i8792|title=Person Page|website=www.thepeerage.com|access-date=2020-12-13}}&lt;/ref&gt; (They attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House and are treated on the [[Social Victorians/People/Colebrooke|Colbrooke page]].)

'''1897 July 2''', Mr. and Mrs. Arthur Paget (#90 in the list of people present) attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did Gerald Gerald Paget

'''1902''', Arthur Henry Fitzroy Paget promoted to rank of General.&lt;ref name=&quot;:2&quot;&gt;&quot;General Rt. Hon. Sir Arthur Henry Fitzroy Paget.&quot; {{Cite web|url=https://www.thepeerage.com/p3392.htm#i33913|title=Person Page|website=www.thepeerage.com|access-date=2020-10-18}}&lt;/ref&gt;

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==

=== Minnie Paget ===
[[File:Mary-Minnie-ne-Stevens-Lady-Paget-as-Cleopatra.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed as Egyptian Cleopatra in an historical costume with fans and a very ornate head-dress|Mary ('Minnie', née Stevens), Lady Paget in costume as Cleopatra. ©National Portrait Gallery, London.]]
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Minnie Paget, Mrs. Arthur Paget, walked in the &quot;Oriental&quot; procession as Cleopatra.&lt;ref name=&quot;:0&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;&lt;ref name=&quot;:4&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; 

John Thomson's portrait of &quot;Mary ('Minnie', née Stevens), Lady Paget as Cleopatra&quot; in costume is photogravure #145 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:1&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;Mrs. Arthur Paget as Cleopatra.&quot;&lt;ref&gt;&quot;Mrs. Arthur Paget as Cleopatra.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158508/Mary-Minnie-ne-Stevens-Lady-Paget-as-Cleopatra.&lt;/ref&gt;

She apparently had a Black attendant, which the ''Morning Post'' described using the n-word, quoted below; offensive language appears in other reports as well, like the description of Minnie Paget's appearance in the report of the American ''Providence [Rhode Island] Evening Telegram''.
*She was dressed in an &quot;Egyptian costume, the train of black crepe de chine embroidered with gold scarabaeus and lined with cloth of gold; skirt of black gauze with lotus flowers worked in gold, and sash of gauze tissue wrought with stones and scarabaeus. The bodice, glittering with gold and diamonds, was held up on the shoulders with straps of large emeralds and diamonds. The square head-dress was of Egyptian cloth of gold, the sphinx-like side pieces being striped black and gold encrusted with diamonds, and in the middle of the forehead hung a large pearl from a ruby; above was the ibis with outstretched wings of diamonds and sapphires, and beyond were peacock feathers standing out, and the back was all looped with pearls and amber. The remainder of the head-dress was of uncut rubies and emeralds, all real stones, surmounted by the jewelled crown of Egypt; round the neck were row upon row of necklaces of various gems, reaching to the waist, and a jewelled girdle fell to the hem; a nigger [sic] held a fan of ostrich feathers over her head.&quot;&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 8, Col. 1b}}
*&quot;Mrs. Arthur Paget appeared in an Egyptian costume, the train being of black crepe de chine embroidered with gold scarabæns, and lined with cloth of gold; skirt of black gauze with lotus flowers worked in gold, and sash of gauze tissue wrought with stones and scarabæns. The bodice, glittering with gold and diamonds, was held up on the shoulders with straps of large emeralds and diamonds.&quot;&lt;ref name=&quot;:5&quot;&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 3b}}
*&quot;Another Cleopatra was Mrs. Arthur Paget, who really looked the character, as she is so dark and Oriental in appearance. Mrs. Paget had a black attendant.&quot;&lt;ref&gt;&quot;Gorgeous Affair. Costume Ball Given by the Duchess of Devonshire in London Last Evening. Many Americans Present. Duchess of Marlborough Appeared as ‘Columbia’ and Depew as Washington.&quot; ''Providence [Rhode Island] Evening Telegram'' Saturday 3 July 1897: 9, Col. 3b [of 8]. ''Google Books''. Retrieved September 2023. https://books.google.com/books?id=gvJeAAAAIBAJ.&lt;/ref&gt;
*&quot;There were also two Cleopatras ..., and Mrs. Arthur Paget looked her character to the life, and her jewels were quite the most magnificent in the room. Mr. Gerald Paget walked beside her, attired very effectively as Mark Antony.&quot;&lt;ref name=&quot;:7&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 32, Col. 2c}}
[[File:Sir-Arthur-Henry-Fitzroy-Paget-as-Edward-the-Black-Prince.jpg|thumb|left|alt=Black-and-white photograph of a standing man richly dressed in armor, with a sword, a cape and a helmet|Sir Arthur Henry Fitzroy Paget in costume as Edward, the Black Prince. ©National Portrait Gallery, London.]][[File:TombaPrincepNegre.JPG|thumb|alt=Closeup of the effigy on the tomb in Canterbury Cathedral showing his armor, helmet and gloves|Effigy of Edward, the Black Prince, Canterbury Cathedral]][[File:Edward, the Black Prince, in Canterbury Cathedral 02.JPG|thumb|alt=Shield of Prince Edward on wall in Canterbury Cathedral|Coat of Arms of Edward, the Black Prince, showing fleurs de lis and lions]]
=== Col. Arthur Paget ===
Arthur Henry Fitzroy Paget (at 91), Col. Arthur Paget, also attended, as Prince Edward of Woodstock, the &quot;Black Prince&quot;:

* &quot;Colonel Arthur Paget assumed the character of Edward the Black Prince in a chain-mail, with black velvet coat embroidered in gold.&quot;&lt;ref name=&quot;:5&quot; /&gt;{{rp|p. 3, Col. 3b}}
* He was dressed as &quot;Edward the Black Prince, in a chain mail, with black velvet coat embroidered in gold, and fur coat worked with lions and fleur-de-lis in gold; black helmet and Prince of Wales's plume.&quot;&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 8, Col. 1b}}

John Thomson's portrait of &quot;Sir Arthur Henry Fitzroy Paget as Edward, the Black Prince&quot; in costume is photogravure #146 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:1&quot; /&gt; The printing on the portrait says, &quot;Colonel Arthur Paget as Edward the Black Prince.&quot;&lt;ref&gt;&quot;Colonel Arthur Paget as Edward the Black Prince.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158509/Sir-Arthur-Henry-Fitzroy-Paget-as-Edward-the-Black-Prince.&lt;/ref&gt;

Edward, the Black Prince (15 June 1330 – 8 June 1376), son of King Edward III, was a successful military leader. If he had not died before his father, he would have been king of England. His tomb is in Canterbury Cathedral with his surcoat, helmet, shield, and gauntlets. A closeup of his effigy in Canterbury Cathedral is at the right, along with his coat of arms, which shows the design on the surcoat or tabard Arthur Paget is wearing.

=== Gerald Paget Paget ===
Gerald Paget Paget (at 237) was dressed as Marc Antony in the Oriental procession (both sources say ''Gerald Paget Paget'').&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt; No obvious candidate for Gerald Paget Paget can be found except for Gerald Cecil Stewart Paget (15 October 1854 – 25 October 1913), Sir Arthur Paget's brother, who seems quite likely, in part because he would be Marc Antony to Minnie Paget's Cleopatra. The ''Gentlewoman'', which calls him Mr. Gerald Paget, says they walked together in the procession.&lt;ref name=&quot;:7&quot; /&gt;{{rp|p. 32, Col. 2c}}

== Demographics ==
*Nationality: Minnie (Mary) Stevens Paget was American, but Arthur Henry Fitzroy Paget was British.

== Family ==

* General Lord Alfred Henry Paget (26 June 1816 – 24 August 1888)&lt;ref&gt;&quot;General Lord Alfred Henry Paget.&quot; {{Cite web|url=https://www.thepeerage.com/p612.htm#i6117|title=Person Page|website=www.thepeerage.com|access-date=2021-11-23}} https://www.thepeerage.com/p612.htm#i6117.&lt;/ref&gt;
* Cecilia Wyndham (baptised 1 November 1829 – 3 May 1914)&lt;ref&gt;&quot;Cecilia Wyndham.&quot; {{Cite web|url=https://www.thepeerage.com/p4699.htm#i46984|title=Person Page|website=www.thepeerage.com|access-date=2021-11-23}} https://www.thepeerage.com/p4699.htm#i46984.&lt;/ref&gt;
*# Victoria Alexandrina Paget (1848 – 2 February 1859)
*# Hon. Evelyn Cecilia Paget (c. 1850 – 17 May 1904)
*# General Rt. Hon. '''Sir Arthur Henry Fitzroy Paget''' (1 March 1851 – 8 December 1928)
*# Admiral Rt. Hon. Sir Alfred Wyndham Paget (26 March 1852 – 17 June 1918)
*# Major George Thomas Cavendish Paget (24 May 1853 – 28 January 1939)
*# Captain '''Gerald Cecil Stewart Paget''' (15 October 1854 – 25 October 1913)
*# Violet Mary Paget (1856 – 13 June 1908)
*# Lt. Sydney Augustus Paget (19 April 1857 – 16 September 1916)
*# Amy Olivia Paget (3 June 1858 – 14 February 1948)
*# Alberta Victoria Paget (1860 – 28 July 1945)
*# '''Almeric Hugh Paget, 1st and last Baron Queenborough''' (14 March 1861 – 22 September 1949)
*# Alice Maud Paget (1863 – 24 December 1925)
*# Alexandra Harriet Paget (1865 – 19 October 1944)
*# Guinevere Eva Paget (1869 – 26 February 1894)


*Arthur Henry Fitzroy Paget (1 March 1851 – 8 December 1928)&lt;ref name=&quot;:2&quot; /&gt;
*Minnie (Mary) Stevens (1853 – May 1919)&lt;ref name=&quot;:3&quot; /&gt;
#Louise Margaret Leila Wemyss Paget ( – 24 September 1958)
#Albert Edward Sydney Louis Paget (23 May 1879 – 2 August 1917)
#Arthur Wyndham Louis Paget (6 March 1888 – 28 February 1966)
#Reginald Scudamore George Paget (6 March 1888 – 11 June 1931)


* Captain Gerald Cecil Stewart Paget (15 October 1854 – 25 October 1913)&lt;ref&gt;&quot;Captain Gerald Cecil Stewart Paget.&quot; {{Cite web|url=https://www.thepeerage.com/p4697.htm#i46970|title=Person Page|website=www.thepeerage.com|access-date=2021-12-01}} https://www.thepeerage.com/p4697.htm#i46970.&lt;/ref&gt;
* Lucy Annie Emily Gardner ( – 15 April 1927)&lt;ref name=&quot;:6&quot; /&gt;
*# Dorothy Cecilia Paget (30 November 1878 – 10 February 1936)
*# Lettice Mina Paget (25 July 1880 – 6 December 1969)

=== Stevens Family ===

* Paran Stevens (11 September 1802 – 25 April 1872)&lt;ref&gt;{{Cite web|url=https://www.findagrave.com/memorial/93965613/paran-stevens|title=Paran Stevens (1802-1872) - Find a Grave Memorial|website=www.findagrave.com|language=en|access-date=2023-09-17}} https://www.findagrave.com/memorial/93965613/paran-stevens.&lt;/ref&gt;
* Eliza Jewett (1 April 1801 – 4 March 1850)&lt;ref&gt;{{Cite web|url=https://www.findagrave.com/memorial/145086867/eliza-stevens|title=Eliza Jewett Stevens (1801-1850) - Find a Grave...|website=www.findagrave.com|language=en|access-date=2023-09-17}} https://www.findagrave.com/memorial/145086867/eliza-stevens.&lt;/ref&gt;
*# Ellen Stevens Melcher (September 1826 – 11 September 1908)&lt;ref&gt;{{Cite web|url=https://www.findagrave.com/memorial/142694577/ellen-melcher|title=Ellen Stevens Melcher (1826-1908) - Find a Grave...|website=www.findagrave.com|language=en|access-date=2023-09-17}} https://www.findagrave.com/memorial/142694577/ellen-melcher.&lt;/ref&gt;
* Marietta Reed (1827 – 3 April 1895)&lt;ref&gt;{{Cite web|url=https://www.findagrave.com/memorial/93965727/marietta-stevens|title=Marietta Reed Stevens (1827-1895) - Find a Grave...|website=www.findagrave.com|language=en|access-date=2023-09-17}} https://www.findagrave.com/memorial/93965727/marietta-stevens.&lt;/ref&gt;
*# Minnie (Mary) Fiske Stevens (13 August 1853 – 20 May 1919)&lt;ref&gt;{{Cite web|url=https://www.findagrave.com/memorial/74543676/mary-fiske-paget|title=Mary Fiske “Minnie” Stevens Paget (1853-1919) -...|website=www.findagrave.com|language=en|access-date=2023-09-17}} https://www.findagrave.com/memorial/74543676/mary-fiske-paget.&lt;/ref&gt;

== Questions and Notes ==
#Arthur Fitzroy Paget was a colonel in 1897, a general in 1902.

== Bibliography ==

# Fitzroy, Sir Almeric. ''Memoirs''. 2 vols. 1925.

== Footnotes ==
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    <title>Maritime Health Research and Education-NET</title>
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==Introduction ==
The Maritime Health Research and Education-NET MAHRE-Net is a non-profit network of researchers, seafarers and other workers, maritime students composed of four parts: 
# Research, based on standardized, health questionnaires.
# Screening for T2 Diabetes mellitus and Hypertension in the fit-for-duty medical examinations
# Health promotion program integrated with the T2DM &amp; HTN screening program
# Systematic Literature Reviews and Reviews of Systematic Reviews.
The primary target study populations include maritime students, seafarers, fishermen, port workers, offshore workers, divers, and their social relations, and other industries. The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.&lt;ref&gt;‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals&lt;/ref&gt;&lt;ref&gt;[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]&lt;/ref&gt; 
We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''&lt;ref&gt; [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]&lt;/ref&gt;,&lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/24737131/&lt;/ref&gt; &lt;ref&gt;https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y&lt;/ref&gt;'''
The research program includes monitoring of the main topics of the EU-Occupational Health strategy: 
# Mental health 
# Ergonomics 
# NCDs Screening for Hypertension and Diabetes Type 2, accurate diagnosis and prevention at the medical examinations
# COVID-19 and other infectious disease
# Safety climate
When they start their practice times at sea, they complete the same questions to identify the influence on their well-being on board. We measure how many of them leave the sea profession and we ask them how they think the profession can continue to be attractive to young people. It is intended to suggest and assist in the implementation of preventive measures based on the results. Later, we continue to ask them at some year intervals with the same questionnaires to assess whether the efforts have helped. We give the same questionnaires to the maritime students in other countries for comparison and learn from their proposals to get the best working conditions. Also, we ask what is needed of teaching to help the industry give them the best condition to stay safe in the job. Different cohort data sources like pre-employment medical health examination data can be tried out for feasibility and validity. A &quot;Cohort&quot; is defined in epidemiological science as a group of people who share the same characteristic, in this case, more or less the same birth years and we take several contacts to them over their life. Cohorts are also started with maritime workers through unions, other organizations, and shipping companies:
==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|International T2 Diabetes Mellitus and Hypertension Research Group]] ==
== [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] ==
==[[Education 2: Supervision of Students Thesis Projects|Education 2: Supervision of Students' Thesis Projects]] ==
==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The Maritime Health Journal Club]]==
==[[Education 4:Effectiveness of training in prevention for type 2 diabetes|Education 4:  Effectiveness of training in prevention for type 2 diabetes]]==
==[[/Standard Questionnaire Based studies/|Questionnaire Based studies: Protocols and Questionnaires]] ==
==[[/Systematic Reviews/|Systematic Review Studies]]  ==
==[[/Organisation / |Organisation]] ==
==[[/Presentations pptx /|Presentations]] ==
==[[/Invitations for collaboration/]] ==
==[[/DRAFT EU Consortium for Maritime Health Research and Education/|Consortium for Maritime Health Research and Education]]  ==
==[[/Pre-diabetes Remission Coaching, Education and Research Network/]]  ==

==[[/ BACKGROUND - open/| Literature background]] ==
== Objectives ==
# International prospective exposure and health risk cohort studies with maritime students and workers 
# All other industries are invited to promote the similar research and education goals  
# Using the standard protocols with different themes 
# Harmonise exposure and outcome information by using standard questionnaires
# Objective and subjective assessments of workplace hazards exposures 
# Develop and validate Job-exposure Matrices 
# Systematic reviews and pooled studies from the cohort rounds
# Cohort-Linkage to pre-entry and follow-up health exams and other health registers
# Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes  Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension]
# International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing  VCT@WORK''']
# Make training materials based on the cohort study and the clinical study outcomes and other scientific sources
# OHS training to maritime doctors, seafarers, fishermen, students, and others 
# Integrate research methodology in the supervision of student’s thesis work
# Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data 
# Keep the Excel data file copies safely (producing country and supervisor)
# Disseminate the knowledge in publications and organize seminars/webinars/symposia
# There is a close relation to the [https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL Maritime Health Portal]

==Contributions to the Industry == 
# The maritime doctors and other maritime health professionals receive updated knowledge on the health risks at sea for the specific jobs and work areas 
# [https://stami.no/slik-skal-stami-sikre-at-kunnskapen-nar-virksomhetene/ Systematic dissemination of knowledge in work environment-e.g. STAMI Specific project]
# Job-Exposure Matrices will support the needed evidence to the health examinations according to the requirements in the ILO/WHO Guidelines
# The companies receive updated knowledge that enables for strategic and thus more cost-effective prevention efforts also in the Job-Exposure Matrices 
# The MAHRE-Net supports the international organizations with the updated scientific evidence for updating the international conventions and regulations 
# The MAHRE-Net supports the Flag states to comply with their obligations to monitor the working and living environments regularly according to ILO Conventions: MLC2006 for seafarers and C188 for fishermen.

==Contributions to Health Risk Prevention == 
The cohort studies can be seen as the diagnostic part of the prevention related to each of the specific items. Guidelines for the prevention for each of the standard questionnaires will be included in the Cohort Protocols
==Phases of the preventive program for single industry==
# Epidemiological standardized questionnaire studies 
# Occupational therapists investigate troubled job positions
# Occupational Medical Doctors supply with their patients from the workplace
# Statistics of work accidents in the specific areas of the workplace years
# Dialogue with companies to improve safety and ergonomics positions
# Improve and continue if they have done well

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education == 
===== For the maritime workers and the industry=====
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the maritime doctors=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Strengths and Weaknesses  == 
In many countries, the interest for a seafaring carrier among young people is rather low. It is a strength to start with the young seafarers at the maritime schools because they can bring fresh perspective and a different way of thinking to the maritime business and help to attract the young seafarers. According to Unicef, most of them are eager to learn, build their experience and apply their skills in the workforce.[https://www.unicef.org/thailand/stories/6-top-benefits-hiring-young-talent Unicef: 6 top benefits hiring young talents ] The method used is easy to implement in a low budget. It is a strength to use the method that immediately identifies risk elements in the work environment that is not seen by the shipping inspectors in the harbors to be amended for the benefit of the seafarers and the companies. In contrast to the register-based studies, these studies identify actual risk elements in the work environment that will never be learned from the register-based studies. Simple frequency analysis is very useful to start preventive work. In addition, the results in graphics are very useful for basic and advanced education. By using standardized short questionnaires, a good response rate is obtained. The General Data Protection Regulation (GDPR) enforced in EU since 2018 can be very complicated to manage in the research. The general survey data we use focuses on the general exposure data in the work environment and not personally identifiable information. Still, an assessment on GDPR data is always needed and in most cases, our survey research do not apply to the same ethical rules as for clinical database research. The students learn how to apply the research methods in their later professional tasks and search the scientific-based knowledge for solving practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for their professional positions as leaders on board. Among the weaknesses is that the response rate might be too low from the start, that they change their mail address so we have no contact and they are not willing to participate or have no time in the later rounds. Another weakness is that the cross-sectional design cannot identify the causal relations in the single studies. However, by comparison of different questionnaire rounds, the health risk hazards might be present in some cohort parts and not in others and thereby contribute to identifying the causal relations.

==The preventive maritime occupational health strategy ==

The maritime occupational and environmental health strategy in development will be based on the need for new knowledge of the main OH indicators. A continued accumulation of new knowledge from MAHRE-Net constitutes the base for continued development of a preventive maritime OH strategy for &quot;constant care&quot; of the workers at the sea.&lt;ref&gt;https://www.maersk.com/about/core-values    &lt;/ref&gt; Based on the national and international OH strategies, e.g. the EU-OSHA  [https://ec.europa.eu/info/law/better-regulation/have-your-say/initiatives/12673-EU-Strategic-Framework-on-Health-and-Safety-at-Work-2021-2027- EU Strategy]  
&lt;ref&gt;https://www.dropbox.com/s/kralxs9yvl569x0/The%20implementation%20of%20Occupational%20Safety%20and%20Health%20in%20the%20EUST_14630_2019_INIT_en.pdf?dl=0&lt;/ref&gt;, The US-CDC and the WHO, ILO OH strategies we selected the four most important OH areas for constant systematic monitoring in the MAHRE-Net: 1. safety-climate, 2. mental health, 3. musculoskeletal risk and 4. Chemical risks. Besides these four main occupational health indicators the following public health indicators can be/are included in each of the surveys: height/weight for Body Mass Index, Smoking, General Health, Alcohol and questions on prevention The validated and international standard questionnaires are available with no extra costs for the student's thesis writing that seems to be an ideal way to make good progress despite the scarce financials. We profit from the results of the outstanding scientists from the Nordic countries and other countries who developed and validated the standardized questionnaires over the latest thirty years. We offer supervision and methodological support for the Bachelor and MScPubHealth graduation thesis and the thesis students in the maritime universities and other health educations like nurses, medical doctors, pharmaceuticals, etc. Data from the Radio Medical services and the seafarer's health examinations are also included. The intention is to obtain a win-win situation with the students getting inspired to continue to do more advanced studies. However, other types of research design apart from the monitor program are encouraged to be made by enthusiastic researchers based on paid clinic time, private time, or funds. Projects with pooling of data from many countries and trends analysis and combining of different questionnaires will require experienced researchers and fundings. One more very important research issue was added in May 2021, to establish early diagnosis of diseases, especially pre-hypertension and pre-diabetes, with evidence that in that time window of disease development, good effect of prevention. Altogether the research activities in MAHRE-Net are intended to be very wide from the most basic levels to the highest levels of competencies with constant education and learning in a preventive perspective.

==[[/ MARITIME HEALTH PORTAL/| Maritime Health Portal]] ==

==Sharing research data==
As a researcher, you are increasingly encouraged, to make your research data available and usable &lt;ref&gt;https://www.elsevier.com/authors/tools-and-resources/research-data&lt;/ref&gt;. However, interviews with researchers revealed a reluctance to share data included a lack of confidence in the utility of the data &lt;ref&gt;https://f1000research.com/articles/7-1641&lt;/ref&gt;Data-sharing is the desired default in the field of public health and a source of much ethical deliberation. Sharing data potentially contributes to the most efficient source of scientific data, but is fraught with contextual challenges which make stakeholders, particularly those in under-resourced contexts hesitant or slow to share &lt;ref&gt; Carrillo-Larco, Rodrigo M., J. Jaime Miranda, and Andre P. Kengne. ‘Data Pooling Efforts in Africa and Latin America’. The Lancet Global Health 5, no. 1 (1 January 2017): e37. https://doi.org/10.1016/S2214-109X(16)30297-2&lt;/ref&gt; 
For example, the Global Body-Mass Index (BMI) Mortality Collaboration1 published their work on BMI as a predictor of all-cause mortality. The investigators pooled individual participant data from 239 prospective studies, with none originating from Latin America or Africa &lt;ref&gt;Body-mass index and all-cause mortality: individual-participant-data meta-analysis of 239 prospective studies in four continents.
Lancet. 2016; 388: 776-786&lt;/ref&gt;, &lt;ref&gt;Anane-Sarpong, Evelyn, Tenzin Wangmo, Claire Leonie Ward, Osman Sankoh, Marcel Tanner, and Bernice Simone Elger. ‘“You Cannot Collect Data Using Your Own Resources and Put It on Open Access”: Perspectives from Africa about Public Health Data-Sharing’. Developing World Bioethics 18, no. 4 (December 2018): 394–405. https://doi.org/10.1111/dewb.12159&lt;/ref&gt; Also collaboration through OMEGA-NET will enhance the scientific output from individual studies and facilitate pooled studies, data sharing, and transfer of tools and skills to make greater and more efficient use of existing cohorts. Researchers from countries outside Europe can participate in COST Actions based on ascertained mutual benefit. Mehlum, Ingrid. &lt;ref&gt;1673f Network on the Coordination and Harmonisation of European Occupational Cohorts (Omega-Net). Occup Environ Med. Vol. 75, 2018. https://doi.org/10.1136/oemed-2018-ICOHabstracts.356.&lt;/ref&gt;

==Links to relevant organizations, documents, and funds == 

[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).


'''Bibliographic References.'''

==References==
{{reflist}}
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    <title>History of Topics in Special Relativity/Three-acceleration</title>
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==History of three-acceleration transformation==

The [[w:Acceleration (special relativity)|Lorentz transformation of three-acceleration]] is given by

:a) &lt;math&gt;\begin{matrix}a_{x}^{\prime}=\frac{a_{x}}{\gamma^{3}\mu^{3}},\ a_{y}^{\prime}=\frac{a_{y}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{y}v}{c^{2}\gamma^{2}\mu^{3}},\ a_{z}^{\prime}=\frac{a_{z}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{z}v}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{u_{x}v}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

or in vector notation in arbitrary directions

:b) &lt;math&gt;\begin{matrix}\mathbf{a}'=\frac{\mathbf{a}}{\gamma^{2}\mu^{2}}-\frac{\mathbf{(a\cdot v)v}\left(\gamma_{v}-1\right)}{v^{2}\gamma^{3}\mu^{3}}+\frac{\mathbf{(a\cdot v)u}}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{\mathbf{v\cdot u}}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

Equations a) were given by [[#Poincaré (1905/06)]], [[#Einstein (1907/08)]], [[#Abraham (1908)]], [[#Laue (1908)]], [[#Brill (1909)]], while the vector notation b) was given by [[#Tamaki (1913)]].

==History==

===Poincaré (1905/06) ===

[[w:Henri Poincaré]] (July 1905, published January 1906) introduces the Lorentz transformation of three-acceleration:&lt;ref group=R&gt;Poincaré (1905/06), p. 160&lt;/ref&gt;

:&lt;math&gt;\frac{d\xi^{\prime}}{dt^{\prime}}=\frac{d\xi}{dt}\frac{1}{k^{3}\mu^{3}},\quad\frac{d\eta^{\prime}}{dt^{\prime}}=\frac{d\eta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\eta\epsilon}{k^{2}\mu^{3}},\quad\frac{d\zeta^{\prime}}{dt^{\prime}}=\frac{d\zeta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\zeta\epsilon}{k^{2}\mu^{3}}&lt;/math&gt; 

where &lt;math&gt;\left(\xi,\ \eta,\ \zeta\right)=\mathbf{u}&lt;/math&gt;, &lt;math&gt;k=\gamma&lt;/math&gt;, &lt;math&gt;\epsilon=v&lt;/math&gt;, &lt;math&gt;\mu=1+\xi\epsilon=1+u_{x}v&lt;/math&gt;.

===Einstein (1907/08) ===

[[w:Albert Einstein]] (December 1907, published 1908) defined the transformation by restricting himself to the x-component (apparently due to a printing error, Einstein's expression misses a cube in the denominator on the right hand side):&lt;ref group=R&gt;Einstein (1907/08), p. 432&lt;/ref&gt;

:&lt;math&gt;\frac{d^{2}x_{0}^{\prime}}{dt^{\prime2}}=\frac{\frac{d}{dt}\left\{ \frac{dx_{0}^{\prime}}{dt'}\right\} }{\beta\left(1-\frac{vx_{0}^{\prime}}{c^{2}}\right)}=\frac{1}{\beta}\frac{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)\ddot{x}_{0}+\left(\dot{x}_{0}-v\right)\frac{v\ddot{x}_{0}}{c^{2}}}{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)}\ \text{etc.}&lt;/math&gt;.

=== Abraham (1908) ===

[[w:Max Abraham]] derived the transformation for three-acceleration by differentiation of the velocity addition in x an y direction:&lt;ref group=R&gt;Abraham (1908), pp. 375-376&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\\
\mathfrak{\dot{q}}_{y}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{y}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{y}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\quad\left(\varkappa=\sqrt{1-\beta^{2}}\right)\\
\mathfrak{\dot{q}}_{z}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{z}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{z}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},
\end{aligned}
&lt;/math&gt;

or simplified by defining vector &lt;math&gt;\mathfrak{\dot{p}}&lt;/math&gt;:

:&lt;math&gt;\begin{matrix}\mathfrak{\dot{p}}=\frac{\mathfrak{\dot{q}}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}\beta\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}}\\
\hline \mathfrak{\dot{q}}_{x}^{\prime}=\mathfrak{\dot{p}}_{x},\ \varkappa\mathfrak{\dot{q}}_{y}^{\prime}=\mathfrak{\dot{p}}_{y},\ \varkappa\mathfrak{\dot{q}}_{z}^{\prime}=\mathfrak{\dot{p}}_{z}
\end{matrix}&lt;/math&gt;

=== Laue (1908) ===

[[w:Max von Laue]] wrote the transformation for three-acceleration as follows:&lt;ref group=R&gt;Laue (1908), p. 840&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{3}\mathfrak{\dot{q}}_{x}, &amp; \mathfrak{\dot{q}}_{y}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{2}\left(\mathfrak{\dot{q}}_{y}+\frac{v\mathfrak{q}_{y}\mathfrak{\dot{q}}_{x}}{c^{2}-v\mathfrak{q}_{x}}\right),\end{aligned}
&lt;/math&gt;

=== Brill (1909) ===

[[w:Alexander von Brill]] wrote the formulas in terms of motions in z-direction:&lt;ref group=R&gt;Brill (1909), p. 210&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{d^{2}x^{\prime}}{dt^{\prime2}}=\frac{d}{dt}\frac{\dot{x}}{k-kq\dot{z}}\cdot\frac{1}{\frac{dt'}{dt}}=\frac{1}{k^{2}}\frac{\ddot{x}(1-q\dot{z})+q\dot{x}\ddot{z}}{(1-q\dot{z})^{3}}\\
\frac{d^{2}z^{\prime}}{dt^{\prime2}}=\frac{d\mathfrak{v}_{z'}^{\prime}}{dt'}=\frac{\ddot{z}\sqrt{1-q^{2}}^{3}}{(1-q\dot{z})^{3}}
\end{matrix}&lt;/math&gt;

=== Tamaki (1913) ===

[[w:Kajuro Tamaki]] was the first to write the formula in three-vector notation:&lt;ref group=R&gt;Tamaki (1913), p. 242&lt;/ref&gt;

:&lt;math&gt;\mathbf{a}'=\frac{\mathbf{a}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]+\frac{1}{\beta}(1-\beta)\mathbf{v}_{1}\left(\mathbf{v}_{1}\mathbf{a}\right)}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}&lt;/math&gt;

and went on to decompose the acceleration into two component vector, one in the direction of &lt;math&gt;\mathbf{v}&lt;/math&gt; and the other perpendicular to it:

:&lt;math&gt;\begin{aligned}\mathbf{a}_{v}^{\prime} &amp; =\frac{\mathbf{a}_{v}-\frac{1}{c^{2}}\beta\left[\mathbf{v}[\mathbf{vq}]\right]_{v}}{\beta^{3}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}\\
\mathbf{a}_{\bar{v}}^{\prime} &amp; =\frac{\mathbf{a}_{\bar{v}}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]_{\bar{v}}}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}
\end{aligned}
&lt;/math&gt;

==References==

{{reflist|3|group=R}}

*{{#section:History of Topics in Special Relativity/relsource|abra08elek}} 
*{{#section:History of Topics in Special Relativity/relsource|brill09}}
*{{#section:History of Topics in Special Relativity/relsource|einst07pri}} 
*{{#section:History of Topics in Special Relativity/relsource|laue08}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}} 
*{{#section:History of Topics in Special Relativity/relsource|tamaki13b}} 

[[Category:History of special relativity]]</text>
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      <id>2622989</id>
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        <username>D.H</username>
        <id>52339</id>
      </contributor>
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==History of three-acceleration transformation==

The [[w:Acceleration (special relativity)|Lorentz transformation of three-acceleration]] is given by

:a) &lt;math&gt;\begin{matrix}a_{x}^{\prime}=\frac{a_{x}}{\gamma^{3}\mu^{3}},\ a_{y}^{\prime}=\frac{a_{y}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{y}v}{c^{2}\gamma^{2}\mu^{3}},\ a_{z}^{\prime}=\frac{a_{z}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{z}v}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{u_{x}v}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

or in vector notation in arbitrary directions

:b) &lt;math&gt;\begin{matrix}\mathbf{a}'=\frac{\mathbf{a}}{\gamma^{2}\mu^{2}}-\frac{\mathbf{(a\cdot v)v}\left(\gamma_{v}-1\right)}{v^{2}\gamma^{3}\mu^{3}}+\frac{\mathbf{(a\cdot v)u}}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{\mathbf{v\cdot u}}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

Equations a) were given by [[#Poincaré (1905/06)]], [[#Einstein (1907/08)]], [[#Abraham (1908)]], [[#Laue (1908)]], [[#Brill (1909)]], while the vector notation b) was given by [[#Tamaki (1913)]].

==History==

===Poincaré (1905/06) ===

[[w:Henri Poincaré]] (July 1905, published January 1906) introduces the Lorentz transformation of three-acceleration:&lt;ref group=R&gt;Poincaré (1905/06), p. 160&lt;/ref&gt;

:&lt;math&gt;\frac{d\xi^{\prime}}{dt^{\prime}}=\frac{d\xi}{dt}\frac{1}{k^{3}\mu^{3}},\quad\frac{d\eta^{\prime}}{dt^{\prime}}=\frac{d\eta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\eta\epsilon}{k^{2}\mu^{3}},\quad\frac{d\zeta^{\prime}}{dt^{\prime}}=\frac{d\zeta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\zeta\epsilon}{k^{2}\mu^{3}}&lt;/math&gt; 

where &lt;math&gt;\left(\xi,\ \eta,\ \zeta\right)=\mathbf{u}&lt;/math&gt;, &lt;math&gt;k=\gamma&lt;/math&gt;, &lt;math&gt;\epsilon=v&lt;/math&gt;, &lt;math&gt;\mu=1+\xi\epsilon=1+u_{x}v&lt;/math&gt;.

===Einstein (1907/08) ===

[[w:Albert Einstein]] (December 1907, published 1908) defined the transformation by restricting himself to the x-component (apparently due to a printing error, Einstein's expression misses a cube in the denominator on the right hand side):&lt;ref group=R&gt;Einstein (1907/08), p. 432&lt;/ref&gt;

:&lt;math&gt;\frac{d^{2}x_{0}^{\prime}}{dt^{\prime2}}=\frac{\frac{d}{dt}\left\{ \frac{dx_{0}^{\prime}}{dt'}\right\} }{\beta\left(1-\frac{vx_{0}^{\prime}}{c^{2}}\right)}=\frac{1}{\beta}\frac{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)\ddot{x}_{0}+\left(\dot{x}_{0}-v\right)\frac{v\ddot{x}_{0}}{c^{2}}}{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)}\ \text{etc.}&lt;/math&gt;.

=== Abraham (1908) ===

[[w:Max Abraham]] derived the transformation for three-acceleration by differentiation of the velocity addition in x an y direction:&lt;ref group=R&gt;Abraham (1908), pp. 375-376&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\\
\mathfrak{\dot{q}}_{y}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{y}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{y}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\quad\left(\varkappa=\sqrt{1-\beta^{2}}\right)\\
\mathfrak{\dot{q}}_{z}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{z}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{z}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},
\end{aligned}
&lt;/math&gt;

or simplified by defining vector &lt;math&gt;\mathfrak{\dot{p}}&lt;/math&gt;:

:&lt;math&gt;\begin{matrix}\mathfrak{\dot{q}}_{x}^{\prime}=\mathfrak{\dot{p}}_{x},\ \varkappa\mathfrak{\dot{q}}_{y}^{\prime}=\mathfrak{\dot{p}}_{y},\ \varkappa\mathfrak{\dot{q}}_{z}^{\prime}=\mathfrak{\dot{p}}_{z}\\
\hline \mathfrak{\dot{p}}=\frac{\mathfrak{\dot{q}}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}\beta\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}}
\end{matrix}&lt;/math&gt;

=== Laue (1908) ===

[[w:Max von Laue]] wrote the transformation for three-acceleration as follows:&lt;ref group=R&gt;Laue (1908), p. 840&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{3}\mathfrak{\dot{q}}_{x}, &amp; \mathfrak{\dot{q}}_{y}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{2}\left(\mathfrak{\dot{q}}_{y}+\frac{v\mathfrak{q}_{y}\mathfrak{\dot{q}}_{x}}{c^{2}-v\mathfrak{q}_{x}}\right),\end{aligned}
&lt;/math&gt;

=== Brill (1909) ===

[[w:Alexander von Brill]] wrote the formulas in terms of motions in z-direction:&lt;ref group=R&gt;Brill (1909), p. 210&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{d^{2}x^{\prime}}{dt^{\prime2}}=\frac{d}{dt}\frac{\dot{x}}{k-kq\dot{z}}\cdot\frac{1}{\frac{dt'}{dt}}=\frac{1}{k^{2}}\frac{\ddot{x}(1-q\dot{z})+q\dot{x}\ddot{z}}{(1-q\dot{z})^{3}}\\
\frac{d^{2}z^{\prime}}{dt^{\prime2}}=\frac{d\mathfrak{v}_{z'}^{\prime}}{dt'}=\frac{\ddot{z}\sqrt{1-q^{2}}^{3}}{(1-q\dot{z})^{3}}
\end{matrix}&lt;/math&gt;

=== Tamaki (1913) ===

[[w:Kajuro Tamaki]] was the first to write the formula in three-vector notation:&lt;ref group=R&gt;Tamaki (1913), p. 242&lt;/ref&gt;

:&lt;math&gt;\mathbf{a}'=\frac{\mathbf{a}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]+\frac{1}{\beta}(1-\beta)\mathbf{v}_{1}\left(\mathbf{v}_{1}\mathbf{a}\right)}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}&lt;/math&gt;

and went on to decompose the acceleration into two component vector, one in the direction of &lt;math&gt;\mathbf{v}&lt;/math&gt; and the other perpendicular to it:

:&lt;math&gt;\begin{aligned}\mathbf{a}_{v}^{\prime} &amp; =\frac{\mathbf{a}_{v}-\frac{1}{c^{2}}\beta\left[\mathbf{v}[\mathbf{vq}]\right]_{v}}{\beta^{3}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}\\
\mathbf{a}_{\bar{v}}^{\prime} &amp; =\frac{\mathbf{a}_{\bar{v}}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]_{\bar{v}}}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}
\end{aligned}
&lt;/math&gt;

==References==

{{reflist|3|group=R}}

*{{#section:History of Topics in Special Relativity/relsource|abra08elek}} 
*{{#section:History of Topics in Special Relativity/relsource|brill09}}
*{{#section:History of Topics in Special Relativity/relsource|einst07pri}} 
*{{#section:History of Topics in Special Relativity/relsource|laue08}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}} 
*{{#section:History of Topics in Special Relativity/relsource|tamaki13b}} 

[[Category:History of special relativity]]</text>
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      <id>2622991</id>
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        <username>D.H</username>
        <id>52339</id>
      </contributor>
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      <model>wikitext</model>
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==History of three-acceleration transformation==

The [[w:Acceleration (special relativity)|Lorentz transformation of three-acceleration]] is given by

:a) &lt;math&gt;\begin{matrix}a_{x}^{\prime}=\frac{a_{x}}{\gamma^{3}\mu^{3}},\ a_{y}^{\prime}=\frac{a_{y}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{y}v}{c^{2}\gamma^{2}\mu^{3}},\ a_{z}^{\prime}=\frac{a_{z}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{z}v}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{u_{x}v}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

or in vector notation in arbitrary directions

:b) &lt;math&gt;\begin{matrix}\mathbf{a}'=\frac{\mathbf{a}}{\gamma^{2}\mu^{2}}-\frac{\mathbf{(a\cdot v)v}\left(\gamma_{v}-1\right)}{v^{2}\gamma^{3}\mu^{3}}+\frac{\mathbf{(a\cdot v)u}}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{\mathbf{v\cdot u}}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

Equations a) were given by [[#Poincaré (1905/06)]], [[#Einstein (1907/08)]], [[#Abraham (1908)]], [[#Laue (1908)]], [[#Brill (1909)]], while the vector notation b) was given by [[#Tamaki (1913)]].

==History==

===Poincaré (1905/06) ===

[[w:Henri Poincaré]] (July 1905, published January 1906) introduces the Lorentz transformation of three-acceleration:&lt;ref group=R&gt;Poincaré (1905/06), p. 160&lt;/ref&gt;

:&lt;math&gt;\frac{d\xi^{\prime}}{dt^{\prime}}=\frac{d\xi}{dt}\frac{1}{k^{3}\mu^{3}},\quad\frac{d\eta^{\prime}}{dt^{\prime}}=\frac{d\eta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\eta\epsilon}{k^{2}\mu^{3}},\quad\frac{d\zeta^{\prime}}{dt^{\prime}}=\frac{d\zeta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\zeta\epsilon}{k^{2}\mu^{3}}&lt;/math&gt; 

where &lt;math&gt;\left(\xi,\ \eta,\ \zeta\right)=\mathbf{u}&lt;/math&gt;, &lt;math&gt;k=\gamma&lt;/math&gt;, &lt;math&gt;\epsilon=v&lt;/math&gt;, &lt;math&gt;\mu=1+\xi\epsilon=1+u_{x}v&lt;/math&gt;.

===Einstein (1907/08) ===

[[w:Albert Einstein]] (December 1907, published 1908) defined the transformation by restricting himself to the x-component (apparently due to a printing error, Einstein's expression misses a cube in the denominator on the right hand side):&lt;ref group=R&gt;Einstein (1907/08), p. 432&lt;/ref&gt;

:&lt;math&gt;\frac{d^{2}x_{0}^{\prime}}{dt^{\prime2}}=\frac{\frac{d}{dt}\left\{ \frac{dx_{0}^{\prime}}{dt'}\right\} }{\beta\left(1-\frac{vx_{0}^{\prime}}{c^{2}}\right)}=\frac{1}{\beta}\frac{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)\ddot{x}_{0}+\left(\dot{x}_{0}-v\right)\frac{v\ddot{x}_{0}}{c^{2}}}{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)}\ \text{etc.}&lt;/math&gt;.

=== Abraham (1908) ===

[[w:Max Abraham]] derived the transformation for three-acceleration by differentiation of the velocity addition in x an y direction:&lt;ref group=R&gt;Abraham (1908), pp. 375-376&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\\
\mathfrak{\dot{q}}_{y}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{y}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{y}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\quad\left(\varkappa=\sqrt{1-\beta^{2}}\right)\\
\mathfrak{\dot{q}}_{z}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{z}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{z}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},
\end{aligned}
&lt;/math&gt;

or simplified by defining vector &lt;math&gt;\mathfrak{\dot{p}}&lt;/math&gt;:

:&lt;math&gt;\begin{matrix}\mathfrak{\dot{q}}_{x}^{\prime}=\mathfrak{\dot{p}}_{x},\qquad\varkappa\mathfrak{\dot{q}}_{y}^{\prime}=\mathfrak{\dot{p}}_{y},\qquad\varkappa\mathfrak{\dot{q}}_{z}^{\prime}=\mathfrak{\dot{p}}_{z}\\
\hline \mathfrak{\dot{p}}=\frac{\mathfrak{\dot{q}}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}\beta\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}}
\end{matrix}&lt;/math&gt;

=== Laue (1908) ===

[[w:Max von Laue]] wrote the transformation for three-acceleration as follows:&lt;ref group=R&gt;Laue (1908), p. 840&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{3}\mathfrak{\dot{q}}_{x}, &amp; \mathfrak{\dot{q}}_{y}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{2}\left(\mathfrak{\dot{q}}_{y}+\frac{v\mathfrak{q}_{y}\mathfrak{\dot{q}}_{x}}{c^{2}-v\mathfrak{q}_{x}}\right),\end{aligned}
&lt;/math&gt;

=== Brill (1909) ===

[[w:Alexander von Brill]] wrote the formulas in terms of motions in z-direction:&lt;ref group=R&gt;Brill (1909), p. 210&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{d^{2}x^{\prime}}{dt^{\prime2}}=\frac{d}{dt}\frac{\dot{x}}{k-kq\dot{z}}\cdot\frac{1}{\frac{dt'}{dt}}=\frac{1}{k^{2}}\frac{\ddot{x}(1-q\dot{z})+q\dot{x}\ddot{z}}{(1-q\dot{z})^{3}}\\
\frac{d^{2}z^{\prime}}{dt^{\prime2}}=\frac{d\mathfrak{v}_{z'}^{\prime}}{dt'}=\frac{\ddot{z}\sqrt{1-q^{2}}^{3}}{(1-q\dot{z})^{3}}
\end{matrix}&lt;/math&gt;

=== Tamaki (1913) ===

[[w:Kajuro Tamaki]] was the first to write the formula in three-vector notation:&lt;ref group=R&gt;Tamaki (1913), p. 242&lt;/ref&gt;

:&lt;math&gt;\mathbf{a}'=\frac{\mathbf{a}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]+\frac{1}{\beta}(1-\beta)\mathbf{v}_{1}\left(\mathbf{v}_{1}\mathbf{a}\right)}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}&lt;/math&gt;

and went on to decompose the acceleration into two component vector, one in the direction of &lt;math&gt;\mathbf{v}&lt;/math&gt; and the other perpendicular to it:

:&lt;math&gt;\begin{aligned}\mathbf{a}_{v}^{\prime} &amp; =\frac{\mathbf{a}_{v}-\frac{1}{c^{2}}\beta\left[\mathbf{v}[\mathbf{vq}]\right]_{v}}{\beta^{3}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}\\
\mathbf{a}_{\bar{v}}^{\prime} &amp; =\frac{\mathbf{a}_{\bar{v}}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]_{\bar{v}}}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}
\end{aligned}
&lt;/math&gt;

==References==

{{reflist|3|group=R}}

*{{#section:History of Topics in Special Relativity/relsource|abra08elek}} 
*{{#section:History of Topics in Special Relativity/relsource|brill09}}
*{{#section:History of Topics in Special Relativity/relsource|einst07pri}} 
*{{#section:History of Topics in Special Relativity/relsource|laue08}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}} 
*{{#section:History of Topics in Special Relativity/relsource|tamaki13b}} 

[[Category:History of special relativity]]</text>
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        <username>D.H</username>
        <id>52339</id>
      </contributor>
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==History of three-acceleration transformation==

The [[w:Acceleration (special relativity)|Lorentz transformation of three-acceleration]] is given by

:a) &lt;math&gt;\begin{matrix}a_{x}^{\prime}=\frac{a_{x}}{\gamma^{3}\mu^{3}},\ a_{y}^{\prime}=\frac{a_{y}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{y}v}{c^{2}\gamma^{2}\mu^{3}},\ a_{z}^{\prime}=\frac{a_{z}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{z}v}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{u_{x}v}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

or in vector notation in arbitrary directions

:b) &lt;math&gt;\begin{matrix}\mathbf{a}'=\frac{\mathbf{a}}{\gamma^{2}\mu^{2}}-\frac{\mathbf{(a\cdot v)v}\left(\gamma_{v}-1\right)}{v^{2}\gamma^{3}\mu^{3}}+\frac{\mathbf{(a\cdot v)u}}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{\mathbf{v\cdot u}}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

Equations a) were given by [[#Poincaré (1905/06)]], [[#Einstein (1907/08)]], [[#Abraham (1908)]], [[#Laue (1908)]], [[#Brill (1909)]], while the vector notation b) was given by [[#Tamaki (1913)]].

==History==

===Poincaré (1905/06) ===

[[w:Henri Poincaré]] (July 1905, published January 1906) introduces the Lorentz transformation of three-acceleration:&lt;ref group=R&gt;Poincaré (1905/06), p. 160&lt;/ref&gt;

:&lt;math&gt;\frac{d\xi^{\prime}}{dt^{\prime}}=\frac{d\xi}{dt}\frac{1}{k^{3}\mu^{3}},\quad\frac{d\eta^{\prime}}{dt^{\prime}}=\frac{d\eta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\eta\epsilon}{k^{2}\mu^{3}},\quad\frac{d\zeta^{\prime}}{dt^{\prime}}=\frac{d\zeta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\zeta\epsilon}{k^{2}\mu^{3}}&lt;/math&gt; 

where &lt;math&gt;\left(\xi,\ \eta,\ \zeta\right)=\mathbf{u}&lt;/math&gt;, &lt;math&gt;k=\gamma&lt;/math&gt;, &lt;math&gt;\epsilon=v&lt;/math&gt;, &lt;math&gt;\mu=1+\xi\epsilon=1+u_{x}v&lt;/math&gt;.

===Einstein (1907/08) ===

[[w:Albert Einstein]] (December 1907, published 1908) defined the transformation by restricting himself to the x-component (apparently due to a printing error, Einstein's expression misses a cube in the denominator on the right hand side):&lt;ref group=R&gt;Einstein (1907/08), p. 432&lt;/ref&gt;

:&lt;math&gt;\frac{d^{2}x_{0}^{\prime}}{dt^{\prime2}}=\frac{\frac{d}{dt}\left\{ \frac{dx_{0}^{\prime}}{dt'}\right\} }{\beta\left(1-\frac{vx_{0}^{\prime}}{c^{2}}\right)}=\frac{1}{\beta}\frac{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)\ddot{x}_{0}+\left(\dot{x}_{0}-v\right)\frac{v\ddot{x}_{0}}{c^{2}}}{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)}\ \text{etc.}&lt;/math&gt;.

=== Abraham (1908) ===

[[w:Max Abraham]] derived the transformation for three-acceleration by differentiation of the velocity addition in x an y direction:&lt;ref group=R&gt;Abraham (1908), pp. 375-376&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\\
\mathfrak{\dot{q}}_{y}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{y}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{y}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\quad\left(\varkappa=\sqrt{1-\beta^{2}}\right)\\
\mathfrak{\dot{q}}_{z}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{z}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{z}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},
\end{aligned}
&lt;/math&gt;

or simplified:

:&lt;math&gt;\begin{matrix}\mathfrak{\dot{q}}_{x}^{\prime}=\mathfrak{\dot{p}}_{x},\qquad\varkappa\mathfrak{\dot{q}}_{y}^{\prime}=\mathfrak{\dot{p}}_{y},\qquad\varkappa\mathfrak{\dot{q}}_{z}^{\prime}=\mathfrak{\dot{p}}_{z}\\
\hline \mathfrak{\dot{p}}=\frac{\mathfrak{\dot{q}}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}\beta\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}}
\end{matrix}&lt;/math&gt;

=== Laue (1908) ===

[[w:Max von Laue]] wrote the transformation for three-acceleration as follows:&lt;ref group=R&gt;Laue (1908), p. 840&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{3}\mathfrak{\dot{q}}_{x}, &amp; \mathfrak{\dot{q}}_{y}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{2}\left(\mathfrak{\dot{q}}_{y}+\frac{v\mathfrak{q}_{y}\mathfrak{\dot{q}}_{x}}{c^{2}-v\mathfrak{q}_{x}}\right),\end{aligned}
&lt;/math&gt;

=== Brill (1909) ===

[[w:Alexander von Brill]] wrote the formulas in terms of motions in z-direction:&lt;ref group=R&gt;Brill (1909), p. 210&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{d^{2}x^{\prime}}{dt^{\prime2}}=\frac{d}{dt}\frac{\dot{x}}{k-kq\dot{z}}\cdot\frac{1}{\frac{dt'}{dt}}=\frac{1}{k^{2}}\frac{\ddot{x}(1-q\dot{z})+q\dot{x}\ddot{z}}{(1-q\dot{z})^{3}}\\
\frac{d^{2}z^{\prime}}{dt^{\prime2}}=\frac{d\mathfrak{v}_{z'}^{\prime}}{dt'}=\frac{\ddot{z}\sqrt{1-q^{2}}^{3}}{(1-q\dot{z})^{3}}
\end{matrix}&lt;/math&gt;

=== Tamaki (1913) ===

[[w:Kajuro Tamaki]] was the first to write the formula in three-vector notation:&lt;ref group=R&gt;Tamaki (1913), p. 242&lt;/ref&gt;

:&lt;math&gt;\mathbf{a}'=\frac{\mathbf{a}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]+\frac{1}{\beta}(1-\beta)\mathbf{v}_{1}\left(\mathbf{v}_{1}\mathbf{a}\right)}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}&lt;/math&gt;

and went on to decompose the acceleration into two component vector, one in the direction of &lt;math&gt;\mathbf{v}&lt;/math&gt; and the other perpendicular to it:

:&lt;math&gt;\begin{aligned}\mathbf{a}_{v}^{\prime} &amp; =\frac{\mathbf{a}_{v}-\frac{1}{c^{2}}\beta\left[\mathbf{v}[\mathbf{vq}]\right]_{v}}{\beta^{3}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}\\
\mathbf{a}_{\bar{v}}^{\prime} &amp; =\frac{\mathbf{a}_{\bar{v}}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]_{\bar{v}}}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}
\end{aligned}
&lt;/math&gt;

==References==

{{reflist|3|group=R}}

*{{#section:History of Topics in Special Relativity/relsource|abra08elek}} 
*{{#section:History of Topics in Special Relativity/relsource|brill09}}
*{{#section:History of Topics in Special Relativity/relsource|einst07pri}} 
*{{#section:History of Topics in Special Relativity/relsource|laue08}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}} 
*{{#section:History of Topics in Special Relativity/relsource|tamaki13b}} 

[[Category:History of special relativity]]</text>
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        <id>52339</id>
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==History of three-acceleration transformation==

The [[w:Acceleration (special relativity)|Lorentz transformation of three-acceleration]] is given by

:a) &lt;math&gt;\begin{matrix}a_{x}^{\prime}=\frac{a_{x}}{\gamma^{3}\mu^{3}},\ a_{y}^{\prime}=\frac{a_{y}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{y}v}{c^{2}\gamma^{2}\mu^{3}},\ a_{z}^{\prime}=\frac{a_{z}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{z}v}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{u_{x}v}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

or in vector notation in arbitrary directions

:b) &lt;math&gt;\begin{matrix}\mathbf{a}'=\frac{\mathbf{a}}{\gamma^{2}\mu^{2}}-\frac{\mathbf{(a\cdot v)v}\left(\gamma_{v}-1\right)}{v^{2}\gamma^{3}\mu^{3}}+\frac{\mathbf{(a\cdot v)u}}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{\mathbf{v\cdot u}}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

Equations a) were given by [[#Poincaré (1905/06)]], [[#Einstein (1907/08)]], [[#Abraham (1908)]], [[#Laue (1908)]], [[#Brill (1909)]], while the vector notation b) was given by [[#Tamaki (1913)]].

==History==

===Poincaré (1905/06) ===

[[w:Henri Poincaré]] (July 1905, published January 1906) introduces the Lorentz transformation of three-acceleration:&lt;ref group=R&gt;Poincaré (1905/06), p. 160&lt;/ref&gt;

:&lt;math&gt;\frac{d\xi^{\prime}}{dt^{\prime}}=\frac{d\xi}{dt}\frac{1}{k^{3}\mu^{3}},\quad\frac{d\eta^{\prime}}{dt^{\prime}}=\frac{d\eta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\eta\epsilon}{k^{2}\mu^{3}},\quad\frac{d\zeta^{\prime}}{dt^{\prime}}=\frac{d\zeta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\zeta\epsilon}{k^{2}\mu^{3}}&lt;/math&gt; 

where &lt;math&gt;\left(\xi,\ \eta,\ \zeta\right)=\mathbf{u}&lt;/math&gt;, &lt;math&gt;k=\gamma&lt;/math&gt;, &lt;math&gt;\epsilon=v&lt;/math&gt;, &lt;math&gt;\mu=1+\xi\epsilon=1+u_{x}v&lt;/math&gt;.

===Einstein (1907/08) ===

[[w:Albert Einstein]] (December 1907, published 1908) defined the transformation by restricting himself to the x-component (apparently due to a printing error, Einstein's expression misses a cube in the denominator on the right hand side):&lt;ref group=R&gt;Einstein (1907/08), p. 432&lt;/ref&gt;

:&lt;math&gt;\frac{d^{2}x_{0}^{\prime}}{dt^{\prime2}}=\frac{\frac{d}{dt}\left\{ \frac{dx_{0}^{\prime}}{dt'}\right\} }{\beta\left(1-\frac{vx_{0}^{\prime}}{c^{2}}\right)}=\frac{1}{\beta}\frac{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)\ddot{x}_{0}+\left(\dot{x}_{0}-v\right)\frac{v\ddot{x}_{0}}{c^{2}}}{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)}\ \text{etc.}&lt;/math&gt;.

=== Abraham (1908) ===

[[w:Max Abraham]] derived the transformation for three-acceleration by differentiation of the velocity addition in x an y direction:&lt;ref group=R&gt;Abraham (1908), pp. 375-376&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\\
\mathfrak{\dot{q}}_{y}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{y}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{y}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\quad\left(\varkappa=\sqrt{1-\beta^{2}}\right)\\
\mathfrak{\dot{q}}_{z}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{z}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{z}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},
\end{aligned}
&lt;/math&gt;

or simplified:

:&lt;math&gt;\begin{matrix}\mathfrak{\dot{q}}_{x}^{\prime}=\mathfrak{\dot{p}}_{x},\qquad\varkappa\mathfrak{\dot{q}}_{y}^{\prime}=\mathfrak{\dot{p}}_{y},\qquad\varkappa\mathfrak{\dot{q}}_{z}^{\prime}=\mathfrak{\dot{p}}_{z}\\
\left(\mathfrak{\dot{p}}=\frac{\mathfrak{\dot{q}}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}\beta\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}}\right)
\end{matrix}&lt;/math&gt;

=== Laue (1908) ===

[[w:Max von Laue]] wrote the transformation for three-acceleration as follows:&lt;ref group=R&gt;Laue (1908), p. 840&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{3}\mathfrak{\dot{q}}_{x}, &amp; \mathfrak{\dot{q}}_{y}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{2}\left(\mathfrak{\dot{q}}_{y}+\frac{v\mathfrak{q}_{y}\mathfrak{\dot{q}}_{x}}{c^{2}-v\mathfrak{q}_{x}}\right),\end{aligned}
&lt;/math&gt;

=== Brill (1909) ===

[[w:Alexander von Brill]] wrote the formulas in terms of motions in z-direction:&lt;ref group=R&gt;Brill (1909), p. 210&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{d^{2}x^{\prime}}{dt^{\prime2}}=\frac{d}{dt}\frac{\dot{x}}{k-kq\dot{z}}\cdot\frac{1}{\frac{dt'}{dt}}=\frac{1}{k^{2}}\frac{\ddot{x}(1-q\dot{z})+q\dot{x}\ddot{z}}{(1-q\dot{z})^{3}}\\
\frac{d^{2}z^{\prime}}{dt^{\prime2}}=\frac{d\mathfrak{v}_{z'}^{\prime}}{dt'}=\frac{\ddot{z}\sqrt{1-q^{2}}^{3}}{(1-q\dot{z})^{3}}
\end{matrix}&lt;/math&gt;

=== Tamaki (1913) ===

[[w:Kajuro Tamaki]] was the first to write the formula in three-vector notation:&lt;ref group=R&gt;Tamaki (1913), p. 242&lt;/ref&gt;

:&lt;math&gt;\mathbf{a}'=\frac{\mathbf{a}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]+\frac{1}{\beta}(1-\beta)\mathbf{v}_{1}\left(\mathbf{v}_{1}\mathbf{a}\right)}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}&lt;/math&gt;

and went on to decompose the acceleration into two component vector, one in the direction of &lt;math&gt;\mathbf{v}&lt;/math&gt; and the other perpendicular to it:

:&lt;math&gt;\begin{aligned}\mathbf{a}_{v}^{\prime} &amp; =\frac{\mathbf{a}_{v}-\frac{1}{c^{2}}\beta\left[\mathbf{v}[\mathbf{vq}]\right]_{v}}{\beta^{3}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}\\
\mathbf{a}_{\bar{v}}^{\prime} &amp; =\frac{\mathbf{a}_{\bar{v}}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]_{\bar{v}}}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}
\end{aligned}
&lt;/math&gt;

==References==

{{reflist|3|group=R}}

*{{#section:History of Topics in Special Relativity/relsource|abra08elek}} 
*{{#section:History of Topics in Special Relativity/relsource|brill09}}
*{{#section:History of Topics in Special Relativity/relsource|einst07pri}} 
*{{#section:History of Topics in Special Relativity/relsource|laue08}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}} 
*{{#section:History of Topics in Special Relativity/relsource|tamaki13b}} 

[[Category:History of special relativity]]</text>
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==History of three-acceleration transformation==

The [[w:Acceleration (special relativity)|Lorentz transformation of three-acceleration]] is given by

:a) &lt;math&gt;\begin{matrix}a_{x}^{\prime}=\frac{a_{x}}{\gamma^{3}\mu^{3}},\ a_{y}^{\prime}=\frac{a_{y}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{y}v}{c^{2}\gamma^{2}\mu^{3}},\ a_{z}^{\prime}=\frac{a_{z}}{\gamma^{2}\mu^{2}}+\frac{a_{x}u_{z}v}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{u_{x}v}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

or in vector notation in arbitrary directions

:b) &lt;math&gt;\begin{matrix}\mathbf{a}'=\frac{\mathbf{a}}{\gamma^{2}\mu^{2}}-\frac{\mathbf{(a\cdot v)v}\left(\gamma_{v}-1\right)}{v^{2}\gamma^{3}\mu^{3}}+\frac{\mathbf{(a\cdot v)u}}{c^{2}\gamma^{2}\mu^{3}}\\
\left[\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mu=1-\frac{\mathbf{v\cdot u}}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

Equations a) were given by [[#Poincaré (1905/06)]], [[#Einstein (1907/08)]], [[#Abraham (1908)]], [[#Laue (1908)]], [[#Brill (1909)]], while the vector notation b) was given by [[#Tamaki (1913)]].

==History==

===Poincaré (1905/06) ===

[[w:Henri Poincaré]] (July 1905, published January 1906) introduces the Lorentz transformation of three-acceleration:&lt;ref group=R&gt;Poincaré (1905/06), p. 160&lt;/ref&gt;

:&lt;math&gt;\frac{d\xi^{\prime}}{dt^{\prime}}=\frac{d\xi}{dt}\frac{1}{k^{3}\mu^{3}},\quad\frac{d\eta^{\prime}}{dt^{\prime}}=\frac{d\eta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\eta\epsilon}{k^{2}\mu^{3}},\quad\frac{d\zeta^{\prime}}{dt^{\prime}}=\frac{d\zeta}{dt}\frac{1}{k^{2}\mu^{2}}-\frac{d\xi}{dt}\frac{\zeta\epsilon}{k^{2}\mu^{3}}&lt;/math&gt; 

where &lt;math&gt;\left(\xi,\ \eta,\ \zeta\right)=\mathbf{u}&lt;/math&gt;, &lt;math&gt;k=\gamma&lt;/math&gt;, &lt;math&gt;\epsilon=v&lt;/math&gt;, &lt;math&gt;\mu=1+\xi\epsilon=1+u_{x}v&lt;/math&gt;.

===Einstein (1907/08) ===

[[w:Albert Einstein]] (December 1907, published 1908) defined the transformation by restricting himself to the x-component (apparently due to a printing error, Einstein's expression misses a cube in the denominator on the right hand side):&lt;ref group=R&gt;Einstein (1907/08), p. 432&lt;/ref&gt;

:&lt;math&gt;\frac{d^{2}x_{0}^{\prime}}{dt^{\prime2}}=\frac{\frac{d}{dt}\left\{ \frac{dx_{0}^{\prime}}{dt'}\right\} }{\beta\left(1-\frac{vx_{0}^{\prime}}{c^{2}}\right)}=\frac{1}{\beta}\frac{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)\ddot{x}_{0}+\left(\dot{x}_{0}-v\right)\frac{v\ddot{x}_{0}}{c^{2}}}{\left(1-\frac{v\dot{x}_{0}}{c^{2}}\right)}\ \text{etc.}&lt;/math&gt;.

=== Abraham (1908) ===

[[w:Max Abraham]] derived the transformation for three-acceleration by differentiation of the velocity addition in x an y direction:&lt;ref group=R&gt;Abraham (1908), pp. 375-376&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\\
\mathfrak{\dot{q}}_{y}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{y}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{y}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},\quad\left(\varkappa=\sqrt{1-\beta^{2}}\right)\\
\mathfrak{\dot{q}}_{z}^{\prime} &amp; =\frac{\mathfrak{\dot{q}}_{z}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}_{z}\beta\mathfrak{\dot{q}}_{x}\varkappa^{2}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}},
\end{aligned}
&lt;/math&gt;

or simplified:

:&lt;math&gt;\begin{matrix}\mathfrak{\dot{q}}_{x}^{\prime}=\mathfrak{\dot{p}}_{x},\qquad\varkappa\mathfrak{\dot{q}}_{y}^{\prime}=\mathfrak{\dot{p}}_{y},\qquad\varkappa\mathfrak{\dot{q}}_{z}^{\prime}=\mathfrak{\dot{p}}_{z}\\
\left(\mathfrak{\dot{p}}=\frac{\mathfrak{\dot{q}}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{2}}+\frac{\mathfrak{q}\beta\mathfrak{\dot{q}}_{x}\varkappa^{3}}{\left(1-\beta\mathfrak{q}_{x}\right)^{3}}\right)
\end{matrix}&lt;/math&gt;

=== Laue (1908) ===

[[w:Max von Laue]] wrote the transformation for three-acceleration as follows:&lt;ref group=R&gt;Laue (1908), p. 840&lt;/ref&gt;

:&lt;math&gt;\begin{aligned}\mathfrak{\dot{q}}_{x}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{3}\mathfrak{\dot{q}}_{x}, &amp; \mathfrak{\dot{q}}_{y}^{\prime} &amp; =\left(\frac{c\sqrt{c^{2}-v^{2}}}{c^{2}-v\mathfrak{q}_{x}}\right)^{2}\left(\mathfrak{\dot{q}}_{y}+\frac{v\mathfrak{q}_{y}\mathfrak{\dot{q}}_{x}}{c^{2}-v\mathfrak{q}_{x}}\right),\end{aligned}
&lt;/math&gt;

=== Brill (1909) ===

[[w:Alexander von Brill]] wrote the formulas in terms of motions in z-direction:&lt;ref group=R&gt;Brill (1909), p. 210&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{d^{2}x^{\prime}}{dt^{\prime2}}=\frac{d}{dt}\frac{\dot{x}}{k-kq\dot{z}}\cdot\frac{1}{\frac{dt'}{dt}}=\frac{1}{k^{2}}\frac{\ddot{x}(1-q\dot{z})+q\dot{x}\ddot{z}}{(1-q\dot{z})^{3}}\\
\frac{d^{2}z^{\prime}}{dt^{\prime2}}=\frac{d\mathfrak{v}_{z'}^{\prime}}{dt'}=\frac{\ddot{z}\sqrt{1-q^{2}}^{3}}{(1-q\dot{z})^{3}}
\end{matrix}&lt;/math&gt;

=== Tamaki (1913) ===

[[w:Kajuro Tamaki]] was the first to write the formula in a single three-vector formula:&lt;ref group=R&gt;Tamaki (1913), p. 242&lt;/ref&gt;

:&lt;math&gt;\mathbf{a}'=\frac{\mathbf{a}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]+\frac{1}{\beta}(1-\beta)\mathbf{v}_{1}\left(\mathbf{v}_{1}\mathbf{a}\right)}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}&lt;/math&gt;

and went on to decompose the acceleration into two component vector, one in the direction of &lt;math&gt;\mathbf{v}&lt;/math&gt; and the other perpendicular to it:

:&lt;math&gt;\begin{aligned}\mathbf{a}_{v}^{\prime} &amp; =\frac{\mathbf{a}_{v}-\frac{1}{c^{2}}\beta\left[\mathbf{v}[\mathbf{vq}]\right]_{v}}{\beta^{3}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}\\
\mathbf{a}_{\bar{v}}^{\prime} &amp; =\frac{\mathbf{a}_{\bar{v}}-\frac{1}{c^{2}}\left[\mathbf{v}[\mathbf{vq}]\right]_{\bar{v}}}{\beta^{2}\left\{ 1-\frac{1}{c^{2}}(\mathbf{vq})\right\} ^{3}}
\end{aligned}
&lt;/math&gt;

==References==

{{reflist|3|group=R}}

*{{#section:History of Topics in Special Relativity/relsource|abra08elek}} 
*{{#section:History of Topics in Special Relativity/relsource|brill09}}
*{{#section:History of Topics in Special Relativity/relsource|einst07pri}} 
*{{#section:History of Topics in Special Relativity/relsource|laue08}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}} 
*{{#section:History of Topics in Special Relativity/relsource|tamaki13b}} 

[[Category:History of special relativity]]</text>
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    <title>Studies of Euler diagrams/transformations</title>
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__NOTOC__
These are pairs of functions in the same clan ([[Equivalence classes of Boolean functions#NP|NP equivalence class]]), so they can be expressed in terms of each other.&lt;br&gt;
&lt;span style=&quot;opacity: .5; font-size: 80%;&quot;&gt;The clan numbers refer to the [[4-ary Boolean functions; clans in rational order|rational ordering]]. (Which will at some point be replaced by a better one.)&lt;/span&gt;

The transformation from one to the other is a {{w|signed permutation}}, which means that arguments are negated and permuted.&lt;br&gt;
&lt;small&gt;It can be just a set of negated places or just a permutation. (These cases are marked with N, P or NP respecitively.)&lt;/small&gt; 


=={{anchor|dakota_tinora}}&lt;span style=&quot;opacity: .5;&quot;&gt;clan 84:&lt;/span&gt; {{boolfname|dakota}} and {{boolflink|tinora}} &amp;nbsp; (NP)==

{{Collapsible START||open full strong}}
This Euler diagram has no symmetry. Therefore the transformation of one into the other is unique.&lt;br&gt;
The one from left to right is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Orange}~3} \\
{\color{Blue}\neg 2} &amp; {\color{Red}~0} &amp; {\color{ForestGreen}\neg 1} &amp; {\color{Orange}~3}
\end{pmatrix}
&lt;/math&gt;. &amp;nbsp; It means the following:
* The old red border becomes the new blue border, and the orientation changes. &lt;small&gt;(The spikes pointed upward, now they point downward.)&lt;/small&gt; 
* The old green border becomes the new red border. &lt;small&gt;(The orientation stays the same.)&lt;/small&gt;
* The old blue border becomes the new green border, and the orientation changes. &lt;small&gt;(The spikes pointed left, now they point right.)&lt;/small&gt;
* The yellow border remains unchanged.

{| style=&quot;text-align: center;&quot;
| [[File:EuDi; dakota.svg|400px|thumb|{{boolfname|dakota}}]]
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; tinora.svg|400px|thumb|{{boolfname|tinora}}]]
|}

The transformation from right to left is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Orange}~3} \\
{\color{ForestGreen}~1} &amp; {\color{Blue}\neg 2} &amp; {\color{Red}\neg 0} &amp; {\color{Orange}~3}
\end{pmatrix}
&lt;/math&gt;, 
the inverse of the one shown above.
{{Studies of Euler diagrams/transformations/dakota tinora/dh}}
{{Collapsible END}}


==={{anchor|dagoro_darimi}}&lt;span style=&quot;opacity: .5;&quot;&gt;clan 157:&lt;/span&gt; {{boolflink|dagoro}} and {{boolflink|darimi}} &amp;nbsp; (NP)===

{{Collapsible START||collapsed full}}
{{boolfname|Dagoro}} is a gap variant of {{boolfname|tinora}} (shown above on the right).

These two functions do not have the same set of (relevant) arguments. But that is not a problem.&lt;br&gt;
One can see {{boolfname|dagoro}} as a 5-ary function with an irrelevant input ''E'', and {{boolfname|darimi}} has an irrelevant input ''A''.&lt;br&gt;

The transformation from left to right is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Orange}~3} &amp; {\color{Brown}~4} \\
{\color{Orange}\neg 3} &amp; {\color{Blue}~2} &amp; {\color{ForestGreen}~1} &amp; {\color{Brown}~4} &amp; {\color{Red}~0}
\end{pmatrix}
&lt;/math&gt;.
It means the following:
* The old red border becomes the new yellow border, and the orientation changes. &lt;small&gt;(The spikes pointed left, now they point right.)&lt;/small&gt;
* The old green border becomes the new blue border.
* The old blue border becomes the new green border.
* The old yellow border becomes the new brown border.
* The inexistent old brown border &quot;becomes&quot; the inexistent new red border. &lt;small&gt;(There could also be a negator in this place.)&lt;/small&gt;

{| style=&quot;text-align: center;&quot;
| [[File:EuDi; dagoro triangle.svg|401px|thumb|{{boolfname|dagoro}}]]
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; darimi triangle.svg|401px|thumb|{{boolfname|darimi}}]]
|}

The transformation from right to left is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Orange}~3} &amp; {\color{Brown}~4} \\
{\color{Brown}~4} &amp; {\color{Blue}~2} &amp; {\color{ForestGreen}~1} &amp; {\color{Red}\neg 0} &amp; {\color{Orange}~3}
\end{pmatrix}
&lt;/math&gt;, 
the inverse of the one shown above.
{{Template:Studies of Euler diagrams/transformations/dagoro darimi/dh}}
{{Collapsible END}}

=={{anchor|tamino_niliko}}&lt;span style=&quot;opacity: .5;&quot;&gt;clan 109:&lt;/span&gt; {{boolfname|tamino}} and {{boolflink|niliko}} &amp;nbsp; (NP)==

{{Collapsible START||open full strong}}
The diagrams in this &lt;abbr title=&quot;equivalence class&quot;&gt;EC&lt;/abbr&gt; are mirror symmetric.
That means that there are two transformations between each pair of functions.&lt;br&gt;
The one used here to get from left to right is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Orange}~3} \\
{\color{Blue}\neg 2} &amp; {\color{Orange}~3} &amp; {\color{ForestGreen}~1} &amp; {\color{Red}~0}
\end{pmatrix}
&lt;/math&gt;.
The one from right to left is the inverse
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Orange}~3} \\
{\color{Orange}~3} &amp; {\color{Blue}~2} &amp; {\color{Red}\neg 0} &amp; {\color{ForestGreen}~1}
\end{pmatrix}
&lt;/math&gt;.

{| style=&quot;text-align: center;&quot;
| [[File:EuDi; tamino.svg|400px|thumb|{{boolfname|tamino}}]]
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; niliko (symmetric).svg|400px|thumb|{{boolfname|niliko}}]]
|}
{{Collapsible END}}


=={{anchor|dukeli_netuno}}&lt;span style=&quot;opacity: .5;&quot;&gt;clan 203:&lt;/span&gt; {{boolflink|dukeli}} and {{boolfname|netuno}} &amp;nbsp; (NP)==

{{Collapsible START||open full strong}}
Each function can be represented by two mirror symmetric Euler diagrams. Here both are shown for {{boolfname|netuno}}.&lt;br&gt;
The transformation from the chosen diagram of {{boolfname|dukeli}} to the one of {{boolfname|netuno}} &lt;u&gt;above&lt;/u&gt; is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Orange}~3} \\
{\color{Red}~0} &amp; {\color{Orange}~3} &amp; {\color{Blue}\neg 2} &amp; {\color{ForestGreen}\neg 1}
\end{pmatrix}
&lt;/math&gt;.
Its inverse is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Orange}~3} \\
{\color{Red}~0} &amp; {\color{Orange}\neg 3} &amp; {\color{Blue}\neg 2} &amp; {\color{ForestGreen}~1}
\end{pmatrix}
&lt;/math&gt;.

{| style=&quot;text-align: center;&quot;
|rowspan=&quot;2&quot;| [[File:EuDi; dukeli.svg|300px|thumb|{{boolfname|dukeli}}]]
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; netuno mirrored.svg|300px|thumb|{{boolfname|netuno}}]]
|-
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; netuno.svg|300px|thumb|{{boolfname|netuno}}]]
|}

The transformation from the chosen diagram of {{boolfname|dukeli}} to the one of {{boolfname|netuno}} &lt;u&gt;below&lt;/u&gt; is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Orange}~3} \\
{\color{ForestGreen}\neg 1} &amp; {\color{Blue}\neg 2} &amp; {\color{Orange}~3} &amp; {\color{Red}~0}
\end{pmatrix}
&lt;/math&gt;.
Its inverse is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Orange}~3} \\
{\color{Orange}~3} &amp; {\color{Red}\neg 0} &amp; {\color{ForestGreen}\neg 1} &amp; {\color{Blue}~2}
\end{pmatrix}
&lt;/math&gt;.

This whole NP equivalence class with 192 functions (represented by 384 diagrams) can be found in [[Studies of Euler diagrams/dukeli NP|{{boolfname|dukeli}} NP]].
{{Studies of Euler diagrams/transformations/dukeli netuno/dh}}
{{Collapsible END}}

=={{anchor|nagini_medusa}}&lt;span style=&quot;opacity: .5;&quot;&gt;clan 349:&lt;/span&gt; {{boolfname|nagini}} and {{boolflink|medusa}} &amp;nbsp; (P)==

{{Collapsible START||open full strong}}
These diagrams have the symmetry of a rectangle, which can be flipped in four ways.&lt;br&gt;
This means that there are four transformations between each pair of functions in it.&lt;br&gt;
The one used here to get from left to right is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}0} &amp; {\color{ForestGreen}1} &amp; {\color{Blue}2} &amp; {\color{Orange}3} \\
{\color{ForestGreen}1} &amp; {\color{Blue}2} &amp; {\color{Orange}3} &amp; {\color{Red}0}
\end{pmatrix}
&lt;/math&gt;.
The one from right to left is the inverse
&lt;math&gt;
\begin{pmatrix}
{\color{Red}0} &amp; {\color{ForestGreen}1} &amp; {\color{Blue}2} &amp; {\color{Orange}3} \\
{\color{Orange}3} &amp; {\color{Red}0} &amp; {\color{ForestGreen}1} &amp; {\color{Blue}2}
\end{pmatrix}
&lt;/math&gt;.

The arguments are only permuted, but not negated. So only the colors change, but not the direction of the spikes.

{| style=&quot;text-align: center;&quot;
| [[File:EuDi; nagini flat.svg|500px|thumb|{{boolfname|nagini}}]]
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; medusa flat (symmetric).svg|500px|thumb|{{boolfname|medusa}}]]
|}
{{Collapsible END}}


=={{anchor|potero_makoto}}&lt;span style=&quot;opacity: .5;&quot;&gt;clan 15:&lt;/span&gt; {{boolflink|potero}} and {{boolfname|makoto}} &amp;nbsp; (NP)==

{{Collapsible START||collapsed full strong}}
Each function can be represented by two mirror symmetric Euler diagrams. Here both are shown for {{boolfname|makoto}}.&lt;br&gt;
The transformation between the chosen diagram of {{boolflink|potero}} and the one of {{boolfname|makoto}} above is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} \\
{\color{ForestGreen}\neg 1} &amp; {\color{Red}\neg 0} &amp; {\color{Blue}~2}
\end{pmatrix}
&lt;/math&gt;. &lt;small&gt;(It is self-inverse, so it works left to right, and back.)&lt;/small&gt;

{| style=&quot;text-align: center;&quot;
|rowspan=&quot;2&quot;| [[File:EuDi; potero (shapes).svg|300px|thumb|{{boolfname|potero}}]]
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; makoto (shapes).svg|300px|thumb|{{boolfname|makoto}}]]
|-
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; makoto mirrored (shapes).svg|300px|thumb|{{boolfname|makoto}}]]
|}

The transformation from the chosen diagram of {{boolflink|potero}} to the one of {{boolfname|makoto}} below is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} \\
{\color{Blue}\neg 2} &amp; {\color{Red}~0} &amp; {\color{ForestGreen}~1}
\end{pmatrix}
&lt;/math&gt;.
Its inverse is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} \\
{\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Red}\neg 0}
\end{pmatrix}
&lt;/math&gt;.
{{Collapsible END}}

==={{anchor|potero_potula}} {{boolfname|potero}} and {{boolflink|potula}} &amp;nbsp; (N)===

{{Collapsible START||collapsed full}}
These two diagrams differ only in the orientation of border ''A''.
The self-inverse transformation between them is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} \\
{\color{Red}\neg 0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} \\
\end{pmatrix}
&lt;/math&gt;.

{| style=&quot;text-align: center;&quot;
| [[File:EuDi; potero (shapes).svg|300px|thumb|{{boolfname|potero}}]]
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; potula (shapes).svg|300px|thumb|{{boolfname|potula}}]]
|}
{{Collapsible END}}

==={{anchor|potula_basori}} {{boolfname|potula}} and {{boolflink|basori}} &amp;nbsp; (P)===

{{Collapsible START||collapsed full}}
The transformation between the two diagrams is
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} &amp; {\color{Orange}~3} &amp; {\color{Brown}~4} \\
{\color{Red}~0} &amp; {\color{Orange}~3} &amp; {\color{Brown}~4} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2}
\end{pmatrix}
&lt;/math&gt;, which is self-inverse.&lt;br&gt;
&lt;small&gt;
From left to right the columns 3 and 4 are irrelevant, and from right to left the columns 1 and 2.&lt;br&gt;
The entries in these columns could be negated and permuted. Counting these leads to 8 possible transformations.&lt;br&gt;
(That is between the two diagrams. Between the two functions it would be 16.)
&lt;/small&gt;

{| style=&quot;text-align: center;&quot;
| [[File:EuDi; potula (shapes).svg|300px|thumb|{{boolfname|potula}}]]
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; basori (shapes).svg|300px|thumb|{{boolfname|basori}}]]
|}
{{Studies of Euler diagrams/transformations/potula basori/dh}}
{{Collapsible END}}

=={{anchor|dobare_dobipi}}&lt;span style=&quot;opacity: .5;&quot;&gt;clan 19:&lt;/span&gt; {{boolflink|dobare}} and {{boolfname|dobipi}} &amp;nbsp; (N)==

{{Collapsible START||open full strong}}
Euler diagrams in this &lt;abbr title=&quot;equivalence class&quot;&gt;EC&lt;/abbr&gt; have 3-fold {{w|Dihedral group|dihedral symmetry}}.
&lt;small&gt;(This is obfuscated by the conventional representation on the right.)&lt;/small&gt;&lt;br&gt;
Thus there are six transformations from one function to another.&lt;br&gt;
The simplest one between these two is to change the orientation of border ''A'', i.e.
&lt;math&gt;
\begin{pmatrix}
{\color{Red}~0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} \\
{\color{Red}\neg 0} &amp; {\color{ForestGreen}~1} &amp; {\color{Blue}~2} \\
\end{pmatrix}
&lt;/math&gt;.
&amp;nbsp; &lt;small&gt;(Compare [[#potero_potula|''potero'' and ''potula'']].)&lt;/small&gt;

{|
| [[File:EuDi; dobare.svg|thumb|x200px|{{boolfname|dobare}}]]
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; dobipi.svg|thumb|x200px|{{boolfname|dobipi}}]]
|}
{{Collapsible END}}

=={{anchor|barita_filtrates}}{{boolflink|barita}} [[Studies of Euler diagrams/filtrates#barita|filtrates]] &amp;nbsp; (P)==

{{Collapsible START||collapsed full strong}}
{|
| [[File:EuDi; barita filtrate sediri and miniri.svg|thumb|x250px|{{boolflink|miniri}} and {{boolflink|sediri}}]]
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; barita filtrate sedofu and gepofu.svg|thumb|x250px|{{boolflink|gepofu}} and {{boolflink|sedofu}}]]
|style=&quot;padding-left: 50px;&quot;| [[File:EuDi; barita filtrate seduki and putuki.svg|thumb|x250px|{{boolflink|putuki}} and {{boolflink|seduki}}]]
|}
{{Collapsible END}}


=={{anchor|bloatless_alternatives}}bloatless alternatives &amp;nbsp; (N)==

{{Collapsible START||collapsed full strong}}

==={{boolflink|demole}}===

{|
| [[File:EuDi; demole bloatless alternative.svg|x140px]]
|style=&quot;padding-left: 50px;&quot;|  [[File:EuDi; demole bloatless.svg|x230px]]
|}

==={{boolflink|futare}}===

{|
| [[File:EuDi; batch 5; 10 redrawn alternative.svg|x170px]]
|style=&quot;padding-left: 50px;&quot;|  [[File:EuDi; batch 5; 10 redrawn.svg|x250px]]
|}

{{Collapsible END}}</text>
      <sha1>ria8wglgotvp5pcbae3acxnmtv2fozu</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2622875</id>
      <parentid>2622545</parentid>
      <timestamp>2024-04-25T14:03:23Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9774" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240210.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20240330.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240425.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming]]

&lt;/br&gt;</text>
      <sha1>kim9tpibmvsx0le5us0z129t36yxvf7</sha1>
    </revision>
  </page>
  <page>
    <title>Workings of gcc and ld in plain view</title>
    <ns>0</ns>
    <id>285384</id>
    <revision>
      <id>2622914</id>
      <parentid>2622782</parentid>
      <timestamp>2024-04-25T23:04:23Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Library Search Path */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3035" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Borrow ([[Media:Borrow.20230701.pdf |pdf]])
* Overflow ([[Media:gcc.Overflow.20240425.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Overview ====

* Static Linking Overview ([[Media:Link.1.StaticOverview.20181120.pdf |pdf]])

* Dynamic Linking Overview ([[Media:Link.2.DynamicOverview.20181120.pdf |pdf]])

* Shared Library Background ([[Media:Link.3.SharedLibrary.20220924.pdf |pdf]])

==== Library Search Path ====

* Library Search Path ([[Media:Link.4.LibrarySearch.20231002.pdf |pdf]])

* Library Search Using -rpath ([[Media:Link.5.LibraryRPATH.20240426.pdf |pdf]])

* Library Search Examples ([[Media:Link.6.LibraryExample.20240415.pdf |pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming]]</text>
      <sha1>8uuq75y5z0yq01ipsxzkaqfhzjkikdh</sha1>
    </revision>
    <revision>
      <id>2622929</id>
      <parentid>2622914</parentid>
      <timestamp>2024-04-26T02:09:31Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Integer Arithmetic */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3035" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Borrow ([[Media:Borrow.20230701.pdf |pdf]])
* Overflow ([[Media:gcc.Overflow.20240426.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Overview ====

* Static Linking Overview ([[Media:Link.1.StaticOverview.20181120.pdf |pdf]])

* Dynamic Linking Overview ([[Media:Link.2.DynamicOverview.20181120.pdf |pdf]])

* Shared Library Background ([[Media:Link.3.SharedLibrary.20220924.pdf |pdf]])

==== Library Search Path ====

* Library Search Path ([[Media:Link.4.LibrarySearch.20231002.pdf |pdf]])

* Library Search Using -rpath ([[Media:Link.5.LibraryRPATH.20240426.pdf |pdf]])

* Library Search Examples ([[Media:Link.6.LibraryExample.20240415.pdf |pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming]]</text>
      <sha1>0m2w1csyd249dk15gxb1zyqrmgbqysr</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:GUS2Wiki</title>
    <ns>4</ns>
    <id>285491</id>
    <revision>
      <id>2622904</id>
      <parentid>2621163</parentid>
      <timestamp>2024-04-25T19:31:36Z</timestamp>
      <contributor>
        <username>Alexis Jazz</username>
        <id>791434</id>
      </contributor>
      <comment>Updating gadget usage statistics from [[Special:GadgetUsage]] ([[phab:T121049]])</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1276" xml:space="preserve">{{#ifexist:Project:GUS2Wiki/top|{{/top}}|This page provides a historical record of [[Special:GadgetUsage]] through its page history. To get the data in CSV format, see wikitext. To customize this message or add categories, create [[/top]].}}
The following data is cached, and was last updated 2024-04-25T07:27:22Z. A maximum of {{PLURAL:5000|one result is|5000 results are}} available in the cache.
{| class=&quot;sortable wikitable&quot;
! Gadget !! data-sort-type=&quot;number&quot; | Number of users !! data-sort-type=&quot;number&quot; | Active users
|-
|CleanDeletions || 75 || 0
|-
|EnhancedTalk || 1352 || 7
|-
|HideFundraisingNotice || 796 || 12
|-
|HotCat || 866 || 10
|-
|LintHint || 94 || 5
|-
|Round Corners || 1145 || 5
|-
|contribsrange || 363 || 3
|-
|dark-mode || 96 || 3
|-
|dark-mode-toggle || 133 || 7
|-
|edittop || 483 || 6
|-
|popups || 837 || 9
|-
|purge || 699 || 10
|-
|sidebartranslate || 528 || 3
|-
|usurper-count || 96 || 3
|}
* [[Special:GadgetUsage]]
* [[m:Meta:GUS2Wiki/Script|GUS2Wiki]]
&lt;!-- data in CSV format:
CleanDeletions,75,0
EnhancedTalk,1352,7
HideFundraisingNotice,796,12
HotCat,866,10
LintHint,94,5
Round Corners,1145,5
contribsrange,363,3
dark-mode,96,3
dark-mode-toggle,133,7
edittop,483,6
popups,837,9
purge,699,10
sidebartranslate,528,3
usurper-count,96,3
--&gt;</text>
      <sha1>0e2hvxvgtiqzdldqb5lgvv1wqcmod00</sha1>
    </revision>
  </page>
  <page>
    <title>Are wikidebates a good thing?</title>
    <ns>0</ns>
    <id>290246</id>
    <revision>
      <id>2622922</id>
      <parentid>2602042</parentid>
      <timestamp>2024-04-26T00:19:43Z</timestamp>
      <contributor>
        <ip>2803:3380:122B:D800:943A:68D2:F71E:6451</ip>
      </contributor>
      <comment>/* Wikidebates are a good thing */ Edit objection for #DebateTools</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="35435" xml:space="preserve">{{Wikidebate}}

Are wikidebates a good thing? Wikidebates are pages that show a binary (yes-no) question or motion, and a hierarchically itemized structure of arguments for, arguments against, objections to them, objections to objections, and so on. What is meant by &quot;good&quot; is left open for the debaters to consider. The guidelines for wikidebates are at [[Wikidebate/Guidelines]].

Contrasts and key concepts: treatise vs. debate, monologue vs. dialogue, argument, argument for, argument against, objection, objection to objection, statement vs. question, atomic argument vs. compound argument.

== Wikidebates are a good thing ==

=== Pro ===

* {{Argument for}} Wikidebates allow to collect responses to arguments, including poor but relatively often occurring arguments.
* {{Argument for}} Unlike a treatise typical of an encyclopedia or an academic article, a hierarchical debate encourages search for objections. In a sense, it is a more honest search for truth than treatise. It helps the person writing the debate discover problems that would be left not discovered in the treatise form, and one only needs to consider this very page to see that.
** {{Objection}} The dialectic process does not need to be exposed to the reader. The author should use the dialectic process to discover the truth and then present the reader with a treatise format showing the truth.
*** {{Objection}} By exposing the dialectic process to the readers, we teach them how to think. We show them how that works. We encourage them to use a similar process to discover errors and weaknesses in their argumentation. The educational value is great.
*** {{Objection}} For some subjects, it may be impossible to arrive at certain truth. Even if one abandons the specific debate format for them, the dialectical argument structure will leak into the treatise format in some way. It seems neater to expose the dialectical nature of certain problems directly than in the indirect treatise way.
* {{Argument for}} Debates are productively used in British politics and are a staple of British intellectual life, including those organized by [[W:Intelligence Squared]] and [[W:Oxford Union]]. That is inconclusive yet suggestive: perhaps this culture has some merits worth examining.
* {{Argument for}} Dialogues were put to good use in philosophy both ancient and modern, including Plato, Descartes, Berkeley, Hume, Galileo, Lakatos and Hofstadter. For more authors, see [[W:Socratic dialogue]]. There is likely to be some wisdom in this practice. This is inconclusive yet suggestive.
** {{Objection}} True. However, a dialogue in philosophy is a much broader phenomenon than the debate in wikidebate. The kinds of nodes in a wikidebate are arguments, and the kind of relations are &quot;X supports Y&quot; and &quot;X refutes or criticizes Y&quot;. Philosophical dialogs are not restricted to arguments. Plato's dialogs featured questions. (They also featured other material.) Therefore, the wide use of dialogue does little to establish that the specific form of argument for, argument against and objection is the core of the wisdom of these dialogues.
*** {{Objection}} Fair enough. However, the structure of argument for, argument against and objection (which is argument against a subargument) is often used in these dialogues. Admittedly, these dialogues often do feature questions as key elements, and Wikidebates, narrowly construed, do not support questions. (Some already do contain questions, but that may be a violation of the format narrowly construed.) Nonetheless, the plentiful presence of the argument structure in these dialogues has some force, as inconclusive as it may be.
** {{Objection}} One should not make inconclusive arguments. These should be dismissed and not allowed into the discussion; they just make it harder to follow.
*** {{Objection}} Even inconclusive arguments or arguments that have some force but not full force should be admitted as long as they have some cognitive or reasoning value. In this case, the argument serves as a form of double check, to point out a certain practice is widely followed. Sure, even bad practices are all too often in widespread use, but one should perhaps be much more cautious about an entirely innovative practice with no precedent than about practice that is widely used.
*** {{Objection}} A benefit of pointing to existing practice is that the reader can have a look at that practice and see for themselves whether it makes sense.
**** {{Objection}} To see whether the practice makes sense, the reader only needs to have a look at the example wikidebates in Wikiversity. There is no need to point to specific examples elsewhere.
***** {{Objection}} Since Wikiversity is edited by amateurs, it is all too likely to show not the best of the genre. To see the potential of the genre and format, one has to look at some of the best that it can offer, not to some of the most amateurish.
*** {{Objection}} The standard of excluding inconclusive arguments would make the debate format impossible. The fact that arguments made do not have the full logical force is the essence of the dialectical nature of the debate. If an argument has full logical force, no more objections are required.
**** {{Objection}} Even fully conclusive arguments can have some objections, invalid ones.
***** {{Objection}} But then the invalid objections would not be allowed, since they are not only inconclusive but outright invalid. The whole to-and-fro structure would need to be eliminated. And for philosophical subjects that often depend on non-shared assumptions, the arguments are necessarily inconclusive.
****** {{Objection}} It just isn't clear why an argument should be made of which the arguer immediately admits it is inconclusive. While it is true we need to allow inconclusive arguments (as per the above argumentation), we do not need to allow arguments that admit to be inconclusive as part of the argument.
******* {{Objection}} Anyone who makes some form of inductive or extrapolative argument in sciences must know, if they are properly trained, that the argument is logically inconclusive; see [[W:Problem of induction]]. The rejection of arguments of which we know from their form that they are logically inconclusive would eliminate coverage of most of empirical science.
******** {{Objection}} That would only be true for the truly problematic induction, by which someone concludes from &quot;for all cases of F that we know of, G was true&quot; that &quot;for all cases of F, G is true&quot;. The consequence should then speak of ''probability'', not certainty.
********* {{Objection}} That would make the language of discussing empirical facts cumbersome. We would need to admit uncertainty in principle all the time. We would no longer be able to use &quot;the Sun will rise tomorrow&quot; as a certain input into discussion, but rather &quot;probably&quot;. Thus, we would need to sprinkle all statements with &quot;probably&quot;, but that would rob the word &quot;probably&quot; of any differentiating power. As said, the standard of eliminating all inconclusiveness is very impractical.
********** {{Objection}} Good points. However, there are levels of inconclusiveness. The extrapolation from philosophical dialogues to wikidebates is a weak form of induction, unlike &quot;the sun will rise tomorrow&quot;. We need to differentiate weaker and stronger cases of induction, forming a whole relational structure. The structure is probably not a simple scale but rather some relatively complex order relation.
*********** {{Objection}} This is getting mathematical. Better stop here.
* {{Argument for}} Debates give fair hearing to multiple points of view. Some things are objectively true and false, but other depend on culturally-dependent assumptions.
** {{Objection}} So let Hitler speak for 5 minutes and then let Jews speak for 5 minutes?
*** {{Objection}} Sure. The vital thing is not to prevent Hitler from speaking but to make sure Jews cannot be silenced.
**** {{Objection}} Hitler, being a persuasive speaker, can convince masses of a bad doctrine.
***** {{Objection}} The masses are not stupid. They are culpable, to blame. Even if Aryan race were superior, would a truly superior race rule by attempted extermination of another ethnic, rather than acting as a ruler race above that ethnic? Unfortunately, humans in general have a great capacity for evil as well as good. It was Christians who, despite their nominal creed, treated the inhabitants of Americans badly. All too many people are liars, pretenders and frauds. Some of the best hopes is to create societies where the nasty human tendencies are kept in check by set up rules and institutions.
****** {{Objection}} What if the masses believed that the international Jew was to blame for bad things?
******* {{Objection}} What were the Slavic people to blame for, except for being of less worth than the master race? Surely not for international capitalism.
****** {{Objection}} Maybe you are a liar, a pretender and fraud.
******* {{Objection}} Don't believe me a single word. Examine each argument as possibly made by a sly sophist. Try to find independent sources of information and reasoning. Don't trust me on authority.
* {{Argument for}} Debates provide a tool for examining of culture and human thought. They help answer the question: what could have been their reasoning? Most human reasoning is inconclusive; the search is not for conclusive reasoning but rather for reasoning that has some force. Conclusive reasoning is ideal, but perfectly logically conclusive reasoning is found only in mathematics, and sound empirically conclusive reasoning is found in science. In investigations of morality, legality, shoulds and oughts, the reasoning is often inconclusive, yet not entirely hopeless and uninformative, and for such a situation, the debate format is a very good fit.
* {{Argument for}} Debates encourage participants to dare to think and to run the risk of being wrong. Since, one is not required to make correct argument but rather interesting arguments. It is the business of the opposition to find flaws in the arguments made.
* {{Argument for}} The interaction between argument, objection and counterobjection is often more interesting and lively than an encyclopedic monologue. That is not about utility but about attractiveness.
* {{Argument for}} A debate can be seen as a form of persuasive writing. It provides objective evidence that the author took some arguments against their position seriously, and which they are. Objections can be conspicuous by their absence, revealing the lack of understanding, insight or erudition on part of the author.
* {{Argument for}} Britannica's procon.org lists 3 pros and 3 cons for each question it considers, without providing a nested rebuttal structure. It can be accused of some of the things as wikidebates: it suggests relativism instead of absolute truth. Wikidebates have the advantage of listing more arguments and listing rebuttals. The model of procon.org is not conclusive, but is suggestive: they must see some merit in what they are doing.
* {{Argument for}} The debate format invites arguments that are incorrect but perhaps contain some elements of truth, and thereby have some informative value. (A similar argument was made by Mill in defense of free speech.) Such arguments can be likened to ore, from which the metal has to be extracted. Thus, a debate can be likened to a mining and metallurgy operation. A similar notion is the adage, don't bite my finger, look where I am pointing.
** {{Objection}} That is an interesting yet inconclusive point. Encouraging debaters to include low-quality arguments and then defend them by saying that there is at east a grain of truth in there and therefore they should be made does not seem obviously wise. One must be able to criticize arguments as incorrect and the don't-bite-my-finger should not be used as a line of defense. (This is similar to Popper's argument that contradictions between theses, arguments and observations, even if perhaps hard to avoid, must be seen as a problem to be corrected and not as the unavoidable good as implied by Hegel. Popper's conjectures and refutations (for sciences) and Lakatos' proofs and refutations (for mathematics) seem to be better concepts than Hegelian thesis, antithesis and synthesis.)
*** {{Objection}} A fair point. However, the idea is more like allowing ore (impure arguments) into debate, but not defending ore against metal extraction, defending arguments against valid criticism.
* {{Argument for}} A fairly marginal yet real benefit of a debate page is that it maps a question to most relevant further reading. Furthermore, the page can state related search terms to help the reader find more relevant further reading. That is of value even if the dialogue itself was poor.
** {{Objection}} This provides a rationale for creating pages that map questions to Wikipedia articles and further reading, not for the debate format.
* {{Argument for}} The debate structure is sometimes implicit or implied in the philosophical literature. Thus, Popper's ''The Open Society and Its Enemies'' can be seen as a Popper's debate with Plato, Hegel, Marx, Heraclitus and other philosophers. Thus, Popper presents thought and arguments by the &quot;enemies&quot; and provides his responses and criticism. While the format is one of a monologue or treatise, the debate structure is apparent. Especially present is the daring to present objectionable ideas that some will find convincing. Thus, Popper runs the risk of spreading dangerous ideas. He probably thinks these are well spread anyway (which seems true enough), and what he is really doing is neutralizing them as best as he can. What wikidebates are doing is make the debate structure explicit. That is inconclusive yet suggestive.
* {{Argument for}} The debate structure is sometimes implicit or implied in scientific literature. Thus, Darwin's Origin of Species contained not only argument in support but also a section where he addressed possible objections or reservations, which are like arguments against, and Darwin's responses are like objections to these arguments against. What wikidebates are doing is make the debate structure explicit. That is inconclusive yet suggestive.
* {{Argument for}} The comment-response structure is well proven in engineering reviews. The comment is like an argument against and the response is like an objection to that argument for the cases where the comment (issue) is rejected. That is inconclusive yet suggestive.

=== Con ===

* {{Argument against}} Wikidebates support relativism, the idea that nothing is true or correct and to every argument there is a counterargument.
** {{Objection}} They may give that impression but the perceptive reader will realize that is not so. The reader will realize about many arguments that they are wrong even without reading the objections. And for unanswered objections, the reader will often be able to tell that they were wrong. The perceptive reader will not think that an unanswered objection has necessarily won.
*** {{Objection}} Many a reader will get the impression that there is some kind of disagreement between two sides of the argument and that there is no obvious winner. It is in part because no position is eventually sustained, and parties hardly ever admit mistakes, in part since there are so many parties. The responses do not have any party identified, so each response may be as if from a different person. The overall impression is one of relativity, and not of a sustained conclusion. This stands in sharp contrast to a mathematical proof.
**** {{Objection}} These are fair points. They are perhaps not entirely damning, but worth considering.
* {{Argument against}} Wikidebates duplicate encyclopedic articles. Thus, for a debate about the existence of God, there is already a Wikipedia article covering philosophical arguments for the idea much better.
** {{Objection}} That is true to some extent. However, an encyclopedic article does not provide a neat itemized structure of arguments and counterarguments. And encyclopedic articles take the stance of search for verified truth, whereas the essence of arguments in a debate is that they are at least moderately interesting and relevant but inconclusive and open to valid criticism, which may then lead to refinement of the arguments to withstand the criticism. One may be reminded of Hegel's dialectics with its thesis, antithesis and synthesis, but equally well of Karl Popper's Conjectures and Refutations. Thus, science and philosophy begin with problems, which lead to tentative solutions, which lead to criticism and modified versions of the solutions or to other solutions, which then leads to further criticism, etc. Thus, there was Ptolemaic astronomy, on which Kepler was an improvement, but then Newton's laws were an improvement on Kepler's laws, and Einstein's relativity is an improvement on Newton.
** {{Objection}} While Wikipedia does cover arguments for and against on some topics, it thereby in part abandons the style of reporting facts about a subject and switches on meta-level. Thus, instead of presenting us with a true statement X traced to sources, it leaves the truth of X undecided and instead presents arguments Y in support and arguments Z in opposition. Thus, it is not so much that wikidebates duplicate Wikipedia as that Wikipedia duplicates Wikidebates.
*** {{Objection}} That is debatable. It is normal encyclopedic practice for philosophical subjects to cover arguments for and against, e.g. for existence of God. Wikipedia is not going to drop that practice any time soon. And it will attract many more editors than Wikiversity, leading to higher-quality content.
**** {{Objection}} The problem of attracting editors is a fair point, but it pertains to Wikiversity platform itself, and not specifically to wikidebates. At a minimum, wikidebates are worth trying since the benefits of the format are undeniable; in the worst case, Wikiversity will have wikidebate pages of poor quality. By hosting original research, Wikiversity gives up on some of the quality aspects of Wikipedia, in an experiment to see how far the wiki technology can be pushed for the purpose. From that standpoint, wikidebates are a meaningful experiment with a sound format, and whether it will attract enough skilled editors remains to be seen. There is already some good content, so things look hopeful.
*** {{Objection}} Likening the progression of arguments in, say, politics and ethics to science may be misleading. Does ethics really make such progress as physics?
**** {{Objection}} Let us leave aside the question whether ethics makes progress. Even if ethics does not make progress, it is all the more important to show ethics as a debate and not only as a treatise.
**** {{Objection}} Perhaps ethics does not make progress in the same way as physics does. But the improved exploration of the idea and argument space is a real intellectual progress. Thus, we have Hume's is-ought distinction, we have the idea of speciesism, and so on. Ethics as a collection of interesting ideas and arguments made progress; it is for the reader to tell which they find most convincing. The reader is helped by having a menu to choose from, to aid their independent thought. Some readers will fall for bad arguments, some won't.
* {{Argument against}} In order to collect all relevant arguments, the debate would have to become very long and hard to follow.
** {{Objection}} The selection of arguments may be a challenge. For some domains of argumentation, the arguments can be sourced from literature, using literature as inclusion criterion. The problem is real but perhaps not intractable.
* {{Argument against}} Without objective, rule-based or algorithmic inclusion criteria, the wikidebates are open to whim of editors. Poor arguments can easily arrive at the top and poor and poorly worded objections can accumulate. Endless objections, objections to objections, etc. can develop without adding any real value.
** {{Objection}} That is a real problem but perhaps not intractable. We will see. Some vague inclusion criteria can be developed, such as: the argument must have some minimum relevance, some minimum plausibility or at least frequency of being used; the wording should be in native English and if it is not, someone should try to reword it.
** {{Objection}} A similar argument perhaps applies to Wikipedia as well, yet Wikipedia often does fine if imperfect job. The precise details of selection of material to include and the order of presentation are far from algorithmic in Wikipedia, as if a job for mindless untalented people focused on correcting spelling mistakes, comma splice and other aspects of writing mechanics.
* {{Argument against}} Wikidebates create the impression that serious issues can be decided by using material fitting into relatively small chunks of text. In science, that is not so. A scientific article is the proper means of persuasion.
** {{Objection}} Many arguments are relatively simple.
*** {{Objection}} Most simple arguments are incorrect. Worse yet, too many simple incorrect arguments are superficially plausible.
**** {{Objection}} Simple and incorrect arguments can often be refuted by simple and correct counter-arguments. The reader of a debate can learn why certain simple and superficially appealing arguments are wrong or inconclusive. While the reader may not learn the truth of the debated matter, they may be able to avoid certain oversimplifications and fallacies related to the matter. Thus, the reader's understanding of the matter may be advanced.
** {{Objection}} That may be true for some subjects, but not for all. Many subjects afford at least some summarizing arguments to be made. Some arguments actually made can be presented by a sentence or a paragraph.
** {{Objection}} In televised debates, there is not much more room for argument presentation than a wikidebate affords either.
*** {{Objection}} Televised debates may be interesting but are no golden standard for doing science and serious logical analysis. They may oversimplify political issues as well. At worst, televised debates can turn into a display of skilled rhetoric, where search for truth suffers. A scientist proper would perhaps prefer the medium of text anyway for the time it affords to carefully choose words.
**** {{Objection}} It would perhaps be best to look at specific debates that turned bad like that, or provide a link that discusses such debates.
***** {{Objection}} Ideally, yes, but for a start, anyone can try to look at some debates and see whether they were really most productive in search for truth.
****** {{Objection}} Whatever the weaknesses of such debates, the audience gets to hear both sides. By contrast, people all too often tend to read and listen to sources they tend to agree with, staying within their own bubble. If nothing else, the debate format bursts such bubbles.
******* {{Objection}} What if one side got poor defenders of it? The debate about the Catholic Church from Intelligence Squared was unfair: there were two heavyweights arguing against the Catholic Church, whereas the speakers in support were relatively weak, not doing the best advocacy available.
******** {{Objection}} Fair point. However, unlike a televised debate, the wikidebate can eventually be expanded by editors who arrive later, and thus can eventually attract good advocates for the side that initially got poor ones. And the wikidebate can accumulate some of the best further reading arguing for and against the motion available, to complement the debate itself.
* {{Argument against}} Wikidebates are redundant to Britannica's '''procon.org''' pages. Britannica's procon.org  is much more professional than wikidebates can ever hope to be, and will receive many more page views: the arguments are well referenced, well chosen, and there is an initial good introduction into each question. The utility of wikidebates is very small.
** {{Objection}} Wikidebates have a richer format by providing a ''nested structure of objections''. Thus, they allow the true dialectic process to truly unfold. Procon.org is like a debate where speakers can make their initial speeches but are not allowed to respond to each other's arguments; that is not the usual debate format. The educational value of the argument-rebuttal-counterrebuttal structure is great. It exposes the Popperian philosophy: a hypothesis may have a falsifier, but the falsifier itself can be subjection to falsification. Thus, in abstract analysis, a theory is hardly ever fully conclusively falsified. (Practically speaking, it is not quite true: some falsifiers are practically conclusive. This is a more confirmation of the dialectic process: one says something interesting and nearly valid, but not entirely valid. If one is only allowed to say certain and true things and correct arguments, one should better stay silent.)
** {{Objection}} Wikidebates can cover ''many more subjects and questions'' than procon.org currently covers, like Wikipedia covers many more subjects than Britannica does.
** {{Objection}} Britannica is much more professional than Wikipedia in many way, yet Wikipedia is a huge success. There is a hope that Wikidebates can also become a success; we need to see how far the wiki format can be pushed for the purpose. If we give up from the start, we will never find out.
* {{Argument against}} Wikidebates will contribute to the spread of some of the most compelling '''demagoguery''' or '''sophism''' (deceptive yet appealing argument) available. Whether the objections will succeed in neutralizing this kind of material is unclear. They will spread some of the most odious philosophy the world has seen.
** {{Objection}} A fair point. However, the odious philosophy is usually already available for anyone who cares to look. Thus, anyone who has a cursory look at Heraclitus learns that war is the father of all things and a great thing, or something of the sort. Anyone who reads Popper will learn about the ideas of the philosophers he is opposing. To neutralize this kind of matter with objections and debate is some of the best things we can do about it. To censor the world's philosophy does not seem to be the solution.
* {{Argument against}} A debate about whether an '''aggressive war''' is good is too likely to accumulate some of the most '''compelling''' arguments for this '''evil''' proposition. Accumulation of the most compelling arguments for both sides is part of the method of the wikidebate; it is currently indicated to include &quot;all arguments&quot;. These arguments do not need to be logically incorrect, merely rest on fundamentally evil assumptions about what is good, just and moral. The result will be a spread of '''pro-war words''' or propaganda in a particularly concentrated form, possibly to be likened to creation of a critical mass of fissile material that can explode. Another analogy is that to argument armament on both sides, arms race. The hope that arguments against and objections will neutralize these pro-war words is not based on any evidence or proof, and may be merely wishful thinking. This accumulation of evil words may lead to '''tangible harm''' in the real world if some people we be mentally defenseless against these words and will be lead to do bad things.
** {{Objection}} If the above is accepted without reservation, it does not mean that the wikidebate format is bad but rather that some topics are better left not covered and left for the readers to investigate. The problem does not seem to rest with the debate format in particular: Wikipedia has an article covering arguments for slavery and if it covered arguments for aggressive war in similar fashion, it would also create a possibly critical mass of fissile material. Thus, the question is whether we should censor and whether we should approach highly morally problematic subjects with an open mind, not whether we should debate. This question can be discussed without blaming the debate format.
*** {{Objection}} The search for arguments for evil propositions is a key element of the debate format. A Wikipedia article on arguments for aggressive war would have that element incorporated.
**** {{Objection}} Fair enough. Still, the objection can be addressed by avoiding certain subjects, while taking advantage of the debate format for other subjects that are not so highly morally problematic and are actually being publicly debated.
** {{Objection}} Those who seek texts arguing in favor of war will easily find them on the Internet, with no objections stated. Thus, a search for &quot;why war is good&quot; finds multiple articles arguing for the motion at length, including the horrible text ''The Benefits of War'' from 19th century by a U.S. admiral. It is not surprising: those in military positions are likely to find a range of rationalizations and a lot of it does not need to be related to defense. By using more search terms, one can find more.
*** {{Objection}} True. However, that will not create concentration of the most compelling material that can be discovered by multiple editors editing a wikidebate. Even novel devious arguments can be added. It may be likened to experimenting with viruses dangerous to humans, viruses of the mind.
**** {{Comment}} There is some force in the above argument but the risk needs to be put in proportion to the totality of base risk already existing, and the potential benefit from allowing us to quote evil words and then object to them.
** {{Objection}} Karl Popper's ''The Open Society'' quotes various German thinkers speaking positively of war. Thus, Popper also creates an accumulation of material. Maybe the material is not the most compelling, but it is at least material from a range of thinkers. Popper runs the risk that someone will find some of that material compelling. He does not even bother to explain to the reader why an aggressive war is bad; rather, he takes that as granted and uses his quotation material to blame Hegel and various German thinkers for creating and spreading a pro-war philosophy. Popper's stance is anti-war, and he does not think to spread pro-war thinking but rather to fight it by exposing it.
*** {{Objection}} Maybe Popper is wrong in his judgment that what he is doing is harmless. The above is interesting but inconclusive.
** {{Objection}} Anyone who pays any attention to history will realize that the belief that aggressive war is good is widespread through societies and eras. Wars of territorial expansions are know from 18th, 19th, 20th and 21th centuries as well as earlier centuries. Anyone who studies the atrocities of Hitler and Stalin can use them as inspiration and yet, we do not give up documenting these atrocities. Unlike Hitler, Stalin is in part seen positively in Russia, the reasoning behind being too obvious: the self-preservation and growth of a large collective entity is more important than human rights. To prevent aggressive wars, we rely above all on the worldly power of countries that can attack other countries engaging in an aggressive war and the worldly power of defensive military alliances; to some extent, we rely on innate moral sense and its interaction with arguments for and against. What a wikidebate does is explicitly document the structure of the argument space for anyone to understand better the minds of perpetrators. Not everyone would figure it out by themselves, but people with good moral sense are unlikely to be convinced that aggressive war is good; rather, they will understand the thinking of their adversaries better. If they find the thinking of their adversaries odious yet hard to argue against, that may reinforce their realization that evil cannot be overcome by mere argument.
** {{Objection}} A wikidebate collects not only the most compelling evil arguments for war but also the most compelling objections to them and most compelling arguments against war. The result may turn some people to evil but it also may turn some people to good. We do not know what the sum total is.
*** {{Objection}} The precautionary principle would tell us that if we do not know whether the action considered does more good than harm, we should not do it.
**** {{Objection}} Good point. But what it would mean is that we are not allowed to try to understand the evil mind and how it is formed from argument structure perspective, or if we are allowed to do so, then only behind a closed door. That creates analytical harm, by preventing our analytical powers from developing.
**** {{Objection}} The precautionary principle is not obviously correct. It says that we should err on the side of non-intervening in so far as creating a new page, a wikidebate, is creating a new intervention in the state of affairs. Thus, if the probability of more good than harm were 70%, a strict precautionary principle would say we should not do it.
** {{Objection}} The risk needs to be put in perspective. The holy texts of world religions including Judaism, Christianity and Islam contain enough God approval for an aggressive war. The additional possible harm created by a wikidebate does not seem particularly big.
*** {{Objection}} Wikidebate should not be making matters even worse, adding more evil verbal material to that already existing.
**** {{Objection}} The wikidebate does not only add the ''evil'' material, it also adds ''good'' material: the objections against the evil arguments. And the wikidebate cannot do so without quoting these evil arguments. Thus, the development of arsenal against evil involves quoting evil so that one can object to it. Admittedly, none of that is conclusive since on some level of analysis, the questions involved are empirical in principle and cannot be properly answered by mere abstract arguments as we try to do here.
** {{Objection}} In Sea-Wolf, Jack London presents the philosophy of Wolf Larsen, a captain who sees no value in human life except to serve his needs and whims and explains why he thinks so. It is not a defense of that kind of philosophy but an implicit criticism of it. This is one more little confirmation that evil ideas and arguments are easily found and authors are not afraid of exposing them. While this is not about war, presenting such arguments can be quite dangerous: those convinced by them may act in private without being caught. London does not seem to think to be convincing putative criminals to be ones.
*** {{Objection}} London may be wrong.

== Further reading ==

* [[/Essays|You are invited to read or contribute reflective essays on this topic at '''Are wikidebates a good thing/Essays''']]
* {{Subpages}}
* [[Wikipedia:Debate]]
* [[Wikipedia:Dialogue]]
* [[Wikipedia:Socratic dialogue]]
* [[Wikipedia:Proofs and Refutations]]
* [[Wikipedia:Sea-Wolf]]
* [[Wikidata:Q2909277|Wikidata:Conjectures and Refutations]]
* [https://www.filosofieonderwijs.be/files/Vd-Leeuw-Philosophical-dialogue.pdf PHILOSOPHICAL DIALOGUE AND THE SEARCH FOR TRUTH] by Karel L. van der Leeu
* [https://iai.tv/articles/how-should-we-do-philosophy-through-dialogue-or-debate-auid-1138 Should We Do Philosophy Through Dialogue or Debate?], iai.tv
* [https://philosophy.stackexchange.com/questions/16267/is-anyone-now-writing-philosophy-in-the-style-of-plato-the-dialogue Is anyone now writing philosophy in the style of Plato - the Dialogue?], philosophy.stackexchange.com</text>
      <sha1>tvo06eb0drqa47e7no2ncqtgwwtdvb9</sha1>
    </revision>
  </page>
  <page>
    <title>Equirectangular projection</title>
    <ns>0</ns>
    <id>291810</id>
    <revision>
      <id>2622866</id>
      <parentid>2621799</parentid>
      <timestamp>2024-04-25T13:18:59Z</timestamp>
      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>/* Snapshot with a Camera */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28275" xml:space="preserve">== Wiki2Reveal ==
This learning resource can be used as [[Wiki2Reveal]] slides in mathematics courses as introduction to [[w:en:Projective geometry|projective geometry]].

* Start '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=Geometry&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=Geometry Wiki2Reveal]''' [[File:Wiki2Reveal Logo.png|35px]]
  

== Introduction == 
In this learning resource the generation of an equirectangular projection is the objective. The learning resource is use-case driven, [https://niebert.github.io/HuginSample/ AFrame Example Durlach] is the first use-case of equirectangular projection. Look around by dragging the direction of view with the mouse with left mouse button pressed.
[[File:Hugin result in aframe.png|center|300px|Equirectangular image used in spheric image in a browser ]]
&lt;center&gt;
[https://niebert.github.io/HuginSample/ Preview of the equirectangular projection in AFrame] - Drag the preview of the equirectangular image with your mouse button pressed.
&lt;/center&gt;

=== Example of the equirectangular image ===  
Preview the [https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg equirectangular JPG-image] and explore the distortion of the image at the top and the bottom of the rectangular JPEG image. 

=== Navigation with multiple equirectangular images ===
The [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html AFrame Navigation example from the river Rhine in Germany] allows to jump from one location at the river rhine to another location at to the equirectangular preview.
 
[[File:river_rhine_spheric_preview.png|center|300px| - Use case of equirectangular projection  - River Rhine example with multiple locations]]

=== Equirectangular Projection for Maps ===
The following image shows an equirectangular projection of the world. The standard parallel is the equator (plate carrée projection)
[[File:Equirectangular projection SW.jpg|center|300px|upright=1.75|Equirectangular projection of the world; the standard parallel is the equator (plate carrée projection).]]

=== Distortion ===
Equirectangular projection with [[w:en:Tissot's indicatrix|Tissot's indicatrix]] of deformation and with the standard parallels lying on the equator. The deformation of a circle into an [[w:en:Ellipse|ellipse]] is visible on different location the world map. 
 
=== Deformation of a Circle - Distortion Indicator ===
The deformation of the circle is an indicator for the distortion of the image.
[[File:Plate Carrée with Tissot's Indicatrices of Distortion.svg|center|300px|upright=1.75|Equirectangular projection with Tissot's indicatrix of deformation and with the standard parallels lying on the equator]]

=== Areas of Interest ===
In the image above the distortion in the planar projection is
* minimal close to the equator and
* maximal at the south pole and north pole.
Rotating circle over the equator (e.g. intersecting with North and South Pole) can be used to have projections with minimal distortion in the area of interest.

=== North Pole and South Pole ===
The strongest distortion can be found at the north pole and south pole in the following true-colour satellite image of the earth. In the equirectangular projection the top horizontal line of pixels represent the single pixel for the north pole on the sphere model of the earth. Similar to that the  south pole as one pixel is stretched out bottom line of pixels in the equirectangular projection.

== Use-Case ==
From a set of standard images of the camera covering a full the from center point of view. In general you cover 360 degree circle with rectangular standard images and you take image with camera for the sky and the floor (preferred without seeing the tripod that might be used for the other images). These set of images are aggregate in one equirectangular image representing a [https://niebert.github.io/HuginSample/ full spheric panorama image], that be be viewed e.g. in [https://aframe.io/sky Aframe] or other panoramic OpenSource viewes, that support equirectangular images. 

=== EQUI2SPH Projection ===
The underlying type of projection is an equirectangular projection EQUI2SPH, that is used e.g in geographical context, where a sphere is projected to rectangular plane on the map. First of all we explore the use-case in [[3D Modelling]] about the spherical use of an spherical panorama image.

== Distortion ==
The projection creates especially at the &quot;North Pole&quot; and the &quot;South Pole&quot; the heaviest distortion in comparison to distances measured on the surface of the sphere. For panoramic views the distortion is just a matter of storage of the spheric pixel information in a rectangular format. On the image of a market place you will see that the panoramic viewes transfer the rectangular images into a natural view where you can look around and explore the location from different angles and with multiple equirectangular images from many locations (see [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html river Rhine example Cologne]).

== Origin of Terminology and History ==
The ''equirectangular projection'' (also called the ''equidistant cylindrical projection''), and which includes the special case of the ''plate carrée projection'' (also called the ''geographic projection'', ''lat/lon projection'', or ''plane chart''), is a simple [[w:en:map projection|map projection]] attributed to [[w:en:Marinus of Tyre|Marinus of Tyre]], who [[w:en:Ptolemy|Ptolemy]] claims invented the projection about AD 100.&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;5–8, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; The projection maps [[w:en:meridian (geography)|meridians]] to vertical straight lines of constant spacing (for meridional intervals of constant spacing), and [[w:en:circle of latitude|circles of latitude]] to horizontal straight lines of constant spacing (for constant intervals of [[w:en:circle of latitude|parallels]]). The projection is neither [[w:en:equal-area map|equal area]] nor [[w:en:conformal map projection|conformal]]. 

=== Implications of Distortion for Navigation ===
Because of the distortions introduced by this projection, it has little use in [[w:en:navigation|navigation]] or [[w:en:cadastral|cadastral]] mapping and finds its main use in [[w:en:thematic map|thematic mapping]]. 

=== Application in global Raster Datasets ===
In particular, the plate carrée has become a standard for global [[w:en:geographic information system|raster datasets]], such as [[w:en:Celestia|Celestia]], [[w:en:NASA World Wind|NASA World Wind]], the [[w:en:USGS|USGS]] [[w:en:Astrogeology Research Program|Astrogeology Research Program]], and [[w:en:Natural Earth|Natural Earth]], because of the particularly simple relationship between the position of an [[w:en:pixel|image pixel]] on the map and its corresponding geographic location on Earth or other spherical solar system bodies. 

=== Application in panoramic photography ===
In addition it is frequently used in panoramic photography to represent a spherical panoramic image.&lt;ref&gt;{{cite web |title=Equirectangular Projection - PanoTools.org Wiki |url=https://wiki.panotools.org/Equirectangular_Projection |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

== Definition - Equirectangular Projection ==
* '''([[/SPH2EQUI/]])''' The forward projection transforms spherical coordinates into planar coordinates of the equirectangular projection. 
* '''([[/EQUI2SPH/]])''' The reverse projection transforms equirectangular coordinates from the plane back onto the sphere. The formulae presume a [[w:en:figure of the Earth|spherical model]] 

=== Spherical - Longitude and Latitude - SPH ===
* Longitude &lt;math&gt;\lambda \in [ -180^\circ , +180^\circ ] &lt;/math&gt;
* Latitude  &lt;math&gt;\phi \in [ -90^\circ , +90^\circ ]  &lt;/math&gt;

=== Visualization ===
A perspective view of the Earth showing how latitude (&lt;math&gt;\phi&lt;/math&gt;) and longitude (&lt;math&gt;\lambda&lt;/math&gt;) are defined on a spherical model. The graticule spacing is 10 degrees.
[[File:latitude and longitude graticule on a sphere.svg|center|350px|A perspective view of the Earth showing how latitude and longitude]]

==== Longitude - SPH ====
Longitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Greenich Meridian as the [[w:en:Prime Meridian|Prime Meridian]] and is ranging to &lt;math&gt;+180^o&lt;/math&gt; eastward and &lt;math&gt;-180^o&lt;/math&gt; westward. The Greek letter &lt;math&gt;\lambda&lt;/math&gt; (lambda)&lt;ref&gt;{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|title=Coordinate Conversion|website=colorado.edu|access-date=14 March 2018|archive-url=https://web.archive.org/web/20090929121405/http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|archive-date=29 September 2009|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;&quot;&lt;math&gt;\lambda&lt;/math&gt; = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign).&quot;&lt;br /&gt;John P. Snyder, ''[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 Map Projections, A Working Manual] {{Webarchive|url=https://web.archive.org/web/20100701103721/http://pubs.er.usgs.gov/usgspubs/pp/pp1395 |date=2010-07-01 }}'', [[w:en:USGS|USGS]] Professional Paper 1395, page ix&lt;/ref&gt; is used to denote the location of a place on Earth east or west of the Prime Meridian

==== Latitude - SPH ====
Latitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Equator and is ranging to &lt;math&gt;+90^o&lt;/math&gt; towards the North Pole and &lt;math&gt;-90^o&lt;/math&gt; towards the South Pole. The Greek letter &lt;math&gt;\phi&lt;/math&gt; or &lt;math&gt;\varphi&lt;/math&gt; (phi) denotes that angle.

==== Mnemonic - Greek Letter - Phi====
There are two different notations of the greek letter &lt;math&gt; \phi &lt;/math&gt; and &lt;math&gt; \varphi &lt;/math&gt;. In this learning resource the notation &lt;math&gt; \phi &lt;/math&gt; is used to indicate that it denotes the angle at circle that intersects with the North Pole and the South Pole.

=== Definition of Spherical Variables ===
The projections are [[w:en:Function_(mathematics)|mathematical function/mappings]]. For definition of these projections the following variables are defined:
*&lt;math&gt;\lambda&lt;/math&gt; is the [[w:en:longitude|longitude]] of the location to project;
* &lt;math&gt;\phi&lt;/math&gt; is the [[w:en:latitude|latitude]] of the location to project;
* &lt;math&gt;\phi_1&lt;/math&gt; are the standard parallels (north and south of the equator) where the scale of the projection is true;
* &lt;math&gt;\phi_0&lt;/math&gt; is the central parallel of the map (e.g. &lt;math&gt;\phi_0 = 0^\circ &lt;/math&gt; equator);
* &lt;math&gt;\lambda_0&lt;/math&gt; is the central meridian of the map;
* &lt;math&gt;R&lt;/math&gt; is the radius of the globe.
Longitude and latitude variables are defined here in terms of radians.

=== Definition of Equirectangular Planar Variables - EQUI ===
* &lt;math&gt;x_e&lt;/math&gt; is the horizontal coordinate of the projected location on the map;
* &lt;math&gt;y_e&lt;/math&gt; is the vertical coordinate of the projected location on the map;

=== Forward Projection - Spherical to Planar - SPH2EQUI ===
&lt;math&gt;\begin{align}
x &amp;= R \cdot (\lambda - \lambda_0) \cdot \cos (\phi_1)\\
y &amp;= R \cdot (\phi - \phi_0)
\end{align}&lt;/math&gt;

=== Special Case - Forward Projection ===
The {{lang|fr|plate carrée}} ([[w:en:French language|French]], for ''flat square''),&lt;ref&gt;{{Cite web |title=Plate Carrée - a simple example |last=Farkas |first=Gábor |work=O’Reilly Online Learning |date= |access-date=31 December 2022 |url= https://www.oreilly.com/library/view/practical-gis/9781787123328/Text/b21938a9-09f7-46fa-b905-58a0a4ed7d8f.xhtml}}&lt;/ref&gt; is the special case where &lt;math&gt;\varphi_1&lt;/math&gt; is zero. This projection maps ''x'' to be the value of the longitude and ''y'' to be the value of the latitude,&lt;ref&gt;{{cite book |url=https://books.google.co.uk/books?id=-FbVI-2tSuYC&amp;pg=PA119 |p=119 |title=Geographic Information Systems and Science |author1=Paul A. Longley |author2=Michael F. Goodchild |author3=David J. Maguire |author4=David W. Rhind |publisher=John Wiley &amp; Sons |year=2005}}&lt;/ref&gt; and therefore is sometimes called the latitude/longitude or lat/lon(g) projection. 

When the &lt;math&gt;\phi_1&lt;/math&gt; is not zero, such as [[w:en:Marinus of Tyre|Marinus]]'s &lt;math&gt;\phi_1=36&lt;/math&gt;,&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;7, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; or [[w:en:Royal Scottish Geographical Society|Ronald Miller]]'s &lt;math&gt;\phi_1=(37.5, 43.5, 50.5)&lt;/math&gt;,&lt;ref&gt;{{cite web |title=Equidistant Cylindrical (Plate Carrée) |url=https://proj.org/operations/projections/eqc.html |website=PROJ coordinate transformation software library |access-date=25 August 2020}}&lt;/ref&gt; the projection can portray particular latitudes of interest at true scale.

=== Remarks - Ellipsoidal Model ===
While a projection with equally spaced parallels is possible for an '''ellipsoidal model''', it would no longer be equidistant because the distance between parallels on an ellipsoid is not constant. More complex formulae can be used to create an equidistant map whose parallels reflect the true spacing.

===Reverse - Planar to Spherical - EQUI2SPH ===
&lt;math&gt;\begin{align}
\lambda &amp;= \frac{x} {R \cdot \cos (\phi_1)} + \lambda_0\\
\phi &amp;= \frac{y} {R} + \phi_0
\end{align}&lt;/math&gt;

=== Alternative names ===
In spherical panorama viewers, usually:

* &lt;math&gt;\lambda&lt;/math&gt; is called &quot;yaw&quot;;&lt;ref&gt;{{cite web |title=Yaw - PanoTools.org Wiki |url=https://wiki.panotools.org/Yaw |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;
* &lt;math&gt;\phi&lt;/math&gt; is called &quot;pitch&quot;;&lt;ref&gt;{{cite web |title=Pitch - PanoTools.org Wiki |url=https://wiki.panotools.org/Pitch |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

where both are defined in degrees.

== Learning Activities ==
The following learning activities address [[w:en:Projective geometry|Projective Geometry]] from a standard snapshot taken with your camera onto an area on the sphere according to the equirectangular projection. Keep in mind to distinguish between the following projective converters:
*  '''(SPH2EQUI / EQUI2SPH):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x_e,y_e)&lt;/math&gt; coordinates of an equirectangular image
*  '''(SPH2IMG / IMG2SPH):''' &lt;math&gt;(x_e,y_e)&lt;/math&gt; equirectangular coordinates &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.
*  '''(EQUI2IMG / IMG2EQUI):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.

=== Angle of View ===
For the learning activities it is important to understand the 
* '''(HAV)''' Horizontal Angle of View in landscape format and 
* '''(VAV)''' Vertical Angle of View in landscape format. 

=== Learning Task - Angle of View ===
The HAV and VAV differ from camera to camera. Explain why the HAV and VAV are relevant for the calculation of IMG2EQUI projection. Use the following figure to explain the requirements for projection.

=== Visualization of Angle of View ===
The following figure depicts besides the Horizontal (HAV) and Vertical Angle of View (VAV) also the Diagonal Angle of View (DAV).

[[File:Angle of view.svg|350px|center|Horizontal, vertical and diagonal angle of view]]

=== Remark - IMG2EQUI Projection ===
The current projection type of the learning activities is the (IMG2EQUI) projection between the &lt;math&gt;(x,y)&lt;/math&gt; coordinates of a standard image  and the distorted &lt;math&gt;(x_e,y_e)&lt;/math&gt;  coordinates of an equirectangular image and vice verca.

=== HAV and VAV as camera specifc properties ===
Due to the fact that different cameras have different angles of view (HAV and VAV) the visible area of the standard image (taken with your smartphone) might vary. The following animation shows different horizontal angles of view (HAV) e.g. for taking a snapshot vertically upwards towards the sky or blue ceiling.

[[File:Horizontal angle of view.gif|350px|center|dynamic visualization of the Horizontal Angle of View]]

=== IMG2EQUI Projection - Polar Regions ===
In this learning step we take pictures with a standard camera or smartphone and want to calculate the ''(Latitude,Longitude)'' coordinates of the sphere from ''(x,y)'' coordinates of standard image created with your smartphone or camera.
In the first learning step we consider the sphere at the polar regions. These regions are the most distorted areas in the equirectangular projections. 


=== IMG2SPH Projection - Take Snapshots for Learning Task ===
* Take your mobile phone and take two snapshots vertical down and vertical up from your floor and from the ceiling. Select an position in your room where the top and the floor has some visible elements (e.g. a lamp, wooden decorative elements,  ...). Alternatively you can create the two images outside with a cloudy sky and objects lying on the ground. 
* select a center of a circle and a radius that fits into both images (maximize the radius of the circle,
* in this module we will project these two images to the polar regions of the sphere within the rectangular coordinate system of the image.

==== Initial State - Image Sky ====
The following image shows the initial state taking a picture from the sky. The camera is located in the center of the red sphere and takes an image of the blue sky or the ceiling in a room. The ceiling is visualized as a blue plane. The learning task is to calculate the angles for the corresponding point on the sphere. 
 
[[File:Circle plane equirectangular.png|350px|Equirectangular Projection - Sky Image - Polar Projection]]

==== Side view with polar image ====
The side view can be used to calculate the angle. 
[[File:Equirectangular polar side view.gif|350px|Equirectangular projection side view - relevant angle is the red angle]]

==== Used Part of Snapshot with a Camera ====
The first image is snapshot with a camera taking a picture vertically upwards to the ceiling (into the sky). The red circle is the projected area from the image (sky/ceiling) onto a part of the equirectangular image visualized in the next section of the learning resource.

[[File:Equirectangular polar image.svg|350px|Equirectangular source image with circle to be projected]]

==== Equirectangular Projection of Circle - Rectangle ====
Explain, why the circle in the source image of the polar region (North Pole) is a rectangle after projection in the equirectangular image.

[[File:Equirectangular polar image projection.svg|350px|Equirectangular source image with circle to be projected]]

==== Learning Task - Function for Coordinate Transformation IMG2EQUI ====
We denote by &lt;math&gt;P_{IMG}(x,y)&lt;/math&gt; the color information of a pixel in the source image (IMG) and with &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt;  the color information of the pixel in the image in the destination format (EQUI). Define the function &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt; by calculating the corresponding &lt;math&gt;(x,y)&lt;/math&gt; coordinates in the IMG source image. This calculation is used to set pixels in the destination format EQUI by setting  
:&lt;math display=&quot;block&quot;&gt;P_{EQUI}(x_e,y_e):=P_{IMG}(x,y).&lt;/math&gt;
'''Remark:''' You need basic knowledge in [[trigonometry]] to perform this learning task.

==== Coordinate System of Graphics ====
[[File:Graphics coordinate system.svg|thumb|Coordinate system in Graphics]]
The coordinate system of an image has a different orientation in y-axis. This is shown in the diagram.
* &lt;math&gt;x_{max}&lt;/math&gt; is the maximal value of the on the x-axis of the image
* &lt;math&gt;y_{max}&lt;/math&gt; is the maximal value of the on the y-axis  the of the image
* The origin of the coordinate system is on the top left. 
Keep this in mind, when you use the coordinate system for your experiments with projections (see also equirectangular projection.

==== Rectangular to Sphere - Projection ====

[[File:Equirectangular ceiling floor.jpg|thumb|Equirectangular projection with marked ceiling and floor]]
Now we use a the standard rectangular image on the right as in input for the equirectangular projection on the sphere and we view the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image in panoramic preview on the sphere]. What can be observed, if you analyze the projected area of red [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg ceiling and a marked green rectangle on the floor].
* Drag the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image upwards to the ceiling] 
* and [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg downwards to floor].

==== Screenshots - Projection on Sphere of the standard Image ====
{| class=&quot;wikitable&quot;
|+ Screenshot of projection
|-
! Ceiling/Sky !! Floor
|-
| [[File:Equirectangular screenshot ceiling.png|320px|Screenshot ceiling/sky of standard image projected on a sphere with equirectangular projection]] || [[File:Equirectangular screenshot floor.png|320px|Screenshot floor of standard image projected on a sphere with equirectangular projection]]
|}

==== Learning Task - Projection of Sky in Graphics Coordinate System ====

[[File:Equirectangular coordinate system sky.svg|thumb|Equirectangular coordinate system sky / ceiling]]

Graphics have an own coordinate system to display points in pixel graphics or geometric objects like lines, polygons and circles in the coordinate system. The diagram shows the coordinate system of the image. Keep in mind that the coordinate of the y-axis has a different orientation than the y-axis in the standard 2D [[w:de:Cartesian coordinate system|Cartesian coordinate system]]. This information is relevant if you experiment with equirectangular projections in [[Projective Geometry Playground]].

Calculate the &lt;math&gt;y_{sky}&lt;/math&gt; for a specific angle of view of your camera.

==== Learning Task - Calculation ====
[[File:Horizontal angle of view.jpg|thumb|Fullscreen image of Door for HAV/VAV calc
[[File:Horizontal angle of view meaure distance.jpg|thumb|Measure distance from camera position to door ]]
ulation]]

Calculate the equirectangular projection for any &lt;math&gt;(x,y)&lt;/math&gt; point in the red circle of the source image IMG into the destination format EQUI with:
* '''LibreOffice - HAV/VAV:''' Take image of a door and calculate the horizontal and vertical angle of view (HAV and VAV)
* '''LibreOffice - Coordinates:''' Create a Spreadsheet document for the calculation of coordinates from a given coordinate in the equirectangular image the corresponding coordinates in the source image of your camera.
* [[Projective Geometry Playground|Javascript and HTML canvas]] - (see [[Projective Geometry Playground]]),
* [[w:en:GNU Octave|Octave]] with [https://gnu-octave.github.io/packages/image/ image-package]&lt;ref&gt; Carnë Draug,
Hartmut Gimpel, Avinoam Kalma (2022) Image Package Octave URL: https://gnu-octave.github.io/packages/image/ (March, 28th, 2024)&lt;/ref&gt;, or 
* Python with [https://github.com/python-pillow/Pillow Image Processing Library pillow]  by Jeffrey A. Clark   
Select an implementation of your choice. LibreOffice has the minimal requirements on programming skill but only coordinate transformation can be performed without a visual output.

=== Learning Task - Sky - North Pole ===
[[File:Sky image for equirectangular projection.jpg|thumb|Sky - equirectangular projection - north pole]]
A sky image can be used to project a circular area in the source image to the rectangular part at the top of the generated equirectangular image. 

Import the equirectangular projection in Aframe to preview the result.

==== Learning Task - Floor - South Pole ====
[[File:Floor sand for equirectangular.jpg|thumb|Floor Image - sand as demo input for equirectangular projection]]Transfer the lesson learned from north pole to the south pole and project the beach image as floor to the south pole. 
* What are differences and similarities between both projections?


== Final Result ==
[[File:Aldara parks.jpg|thumb|[https://niebert.github.io/HuginSample/Aldara_parks.html Equirectangular Image from Wikiversity used for Aframe 360 Degree Image] (see [[3D Modelling/Create 3D Models/Hugin|Hugin]])]]
{{PanoViewer|Frary Dining Hall 360-degree view.jpg|Dining Hall 360-degree view - with PanoViewer Template}}
* [https://niebert.github.io/HuginSample/ Final Results]  would can be previewed in Aframe (see [https://www.github.com/HuginSample HuginSample Files on Github]) or with the Wiki,
* [https://niebert.github.com/HuginSample/Aldara_parks.html Aldara Parks 360 Degree Image in Aframe] with an already uploaded equirectangular image in WikiMedia Commons by U.Bardins
* [https://panoviewer.toolforge.org/#Aldara_parks.jpg PanoViewer 360 degree view of Aldara Parks image]]

==See also==
* [[Projective Geometry Playground]]
* [[GeoGebra/Perspective Drawing on Mirror|Perspective Drawing on Mirror]]
* [[3D Modelling]]
* [[w:en:Cartography|Cartography]]
* [[w:en:Cassini projection|Cassini projection]]
* [[w:en:Gall–Peters projection|Gall–Peters projection]] with resolution regarding the use of rectangular world maps
* [[w:en:List of map projections|List of map projections]]
* [[w:en:Mercator projection|Mercator projection]]
* [[w:en:360 video projection|360 video projection]]
* [[3D_Modelling/Examples/Panorama_360|3D Modelling - 360 degree panorama]]
* [[Geometry]]
* [[w:en:Projective_geometry|Projective Geometry]]
* [[Portal:Mathematics]]
* [[w:en:GNU Octave|Octave]]
* [[Geogebra]]
* [[b:en:Javascript|Wikibook: Javascript]]
* [[Portal:Mathematics]]

==References==
{{Reflist}}

==External links==
* [https://visibleearth.nasa.gov/view.php?id=57730 Global MODIS based satellite map] The blue marble: land surface, ocean color, and sea ice.
* [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net.
* [http://wiki.panotools.org/Equirectangular Panoramic Equirectangular Projection], PanoTools wiki.
* [https://proj4.org/operations/projections/eqc.html Equidistant Cylindrical (Plate Carrée) in proj4]

== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/3D%20Modelling 3D Modelling]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Equirectangular%20projection https://en.wikiversity.org/wiki/Equirectangular%20projection]
--&gt;
* [https://en.wikiversity.org/wiki/Equirectangular%20projection This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
* Source: Wikiversity  https://en.wikiversity.org/wiki/Equirectangular%20projection
* see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal].
&lt;!-- * Next contents of the course are [[]] --&gt;;

=== Wikipedia2Wikiversiy===
This page was based on the following [https://en.wikipedia.org/wiki/Equirectangular%20projection wikipedia-source page]:
* [https://en.wikipedia.org/wiki/Equirectangular%20projection Equirectangular projection] https://en.wikipedia.org/wiki/Equirectangular%20projection
* Datum: 1/9/2023
* [https://niebert.github.io/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.io/Wikipedia2Wikiversity


[[Category:Map projections]]
[[Category:Equidistant projections]]
[[Category:Cylindrical projections]]
[[Category:Wiki2Reveal]]</text>
      <sha1>mwaiq2y2r9623kh85jbmwdl55wkhyws</sha1>
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      <text bytes="28276" xml:space="preserve">== Wiki2Reveal ==
This learning resource can be used as [[Wiki2Reveal]] slides in mathematics courses as introduction to [[w:en:Projective geometry|projective geometry]].

* Start '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=Geometry&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=Geometry Wiki2Reveal]''' [[File:Wiki2Reveal Logo.png|35px]]
  

== Introduction == 
In this learning resource the generation of an equirectangular projection is the objective. The learning resource is use-case driven, [https://niebert.github.io/HuginSample/ AFrame Example Durlach] is the first use-case of equirectangular projection. Look around by dragging the direction of view with the mouse with left mouse button pressed.
[[File:Hugin result in aframe.png|center|300px|Equirectangular image used in spheric image in a browser ]]
&lt;center&gt;
[https://niebert.github.io/HuginSample/ Preview of the equirectangular projection in AFrame] - Drag the preview of the equirectangular image with your mouse button pressed.
&lt;/center&gt;

=== Example of the equirectangular image ===  
Preview the [https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg equirectangular JPG-image] and explore the distortion of the image at the top and the bottom of the rectangular JPEG image. 

=== Navigation with multiple equirectangular images ===
The [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html AFrame Navigation example from the river Rhine in Germany] allows to jump from one location at the river rhine to another location at to the equirectangular preview.
 
[[File:river_rhine_spheric_preview.png|center|300px| - Use case of equirectangular projection  - River Rhine example with multiple locations]]

=== Equirectangular Projection for Maps ===
The following image shows an equirectangular projection of the world. The standard parallel is the equator (plate carrée projection)
[[File:Equirectangular projection SW.jpg|center|300px|upright=1.75|Equirectangular projection of the world; the standard parallel is the equator (plate carrée projection).]]

=== Distortion ===
Equirectangular projection with [[w:en:Tissot's indicatrix|Tissot's indicatrix]] of deformation and with the standard parallels lying on the equator. The deformation of a circle into an [[w:en:Ellipse|ellipse]] is visible on different location the world map. 
 
=== Deformation of a Circle - Distortion Indicator ===
The deformation of the circle is an indicator for the distortion of the image.
[[File:Plate Carrée with Tissot's Indicatrices of Distortion.svg|center|300px|upright=1.75|Equirectangular projection with Tissot's indicatrix of deformation and with the standard parallels lying on the equator]]

=== Areas of Interest ===
In the image above the distortion in the planar projection is
* minimal close to the equator and
* maximal at the south pole and north pole.
Rotating circle over the equator (e.g. intersecting with North and South Pole) can be used to have projections with minimal distortion in the area of interest.

=== North Pole and South Pole ===
The strongest distortion can be found at the north pole and south pole in the following true-colour satellite image of the earth. In the equirectangular projection the top horizontal line of pixels represent the single pixel for the north pole on the sphere model of the earth. Similar to that the  south pole as one pixel is stretched out bottom line of pixels in the equirectangular projection.

== Use-Case ==
From a set of standard images of the camera covering a full the from center point of view. In general you cover 360 degree circle with rectangular standard images and you take image with camera for the sky and the floor (preferred without seeing the tripod that might be used for the other images). These set of images are aggregate in one equirectangular image representing a [https://niebert.github.io/HuginSample/ full spheric panorama image], that be be viewed e.g. in [https://aframe.io/sky Aframe] or other panoramic OpenSource viewes, that support equirectangular images. 

=== EQUI2SPH Projection ===
The underlying type of projection is an equirectangular projection EQUI2SPH, that is used e.g in geographical context, where a sphere is projected to rectangular plane on the map. First of all we explore the use-case in [[3D Modelling]] about the spherical use of an spherical panorama image.

== Distortion ==
The projection creates especially at the &quot;North Pole&quot; and the &quot;South Pole&quot; the heaviest distortion in comparison to distances measured on the surface of the sphere. For panoramic views the distortion is just a matter of storage of the spheric pixel information in a rectangular format. On the image of a market place you will see that the panoramic viewes transfer the rectangular images into a natural view where you can look around and explore the location from different angles and with multiple equirectangular images from many locations (see [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html river Rhine example Cologne]).

== Origin of Terminology and History ==
The ''equirectangular projection'' (also called the ''equidistant cylindrical projection''), and which includes the special case of the ''plate carrée projection'' (also called the ''geographic projection'', ''lat/lon projection'', or ''plane chart''), is a simple [[w:en:map projection|map projection]] attributed to [[w:en:Marinus of Tyre|Marinus of Tyre]], who [[w:en:Ptolemy|Ptolemy]] claims invented the projection about AD 100.&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;5–8, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; The projection maps [[w:en:meridian (geography)|meridians]] to vertical straight lines of constant spacing (for meridional intervals of constant spacing), and [[w:en:circle of latitude|circles of latitude]] to horizontal straight lines of constant spacing (for constant intervals of [[w:en:circle of latitude|parallels]]). The projection is neither [[w:en:equal-area map|equal area]] nor [[w:en:conformal map projection|conformal]]. 

=== Implications of Distortion for Navigation ===
Because of the distortions introduced by this projection, it has little use in [[w:en:navigation|navigation]] or [[w:en:cadastral|cadastral]] mapping and finds its main use in [[w:en:thematic map|thematic mapping]]. 

=== Application in global Raster Datasets ===
In particular, the plate carrée has become a standard for global [[w:en:geographic information system|raster datasets]], such as [[w:en:Celestia|Celestia]], [[w:en:NASA World Wind|NASA World Wind]], the [[w:en:USGS|USGS]] [[w:en:Astrogeology Research Program|Astrogeology Research Program]], and [[w:en:Natural Earth|Natural Earth]], because of the particularly simple relationship between the position of an [[w:en:pixel|image pixel]] on the map and its corresponding geographic location on Earth or other spherical solar system bodies. 

=== Application in panoramic photography ===
In addition it is frequently used in panoramic photography to represent a spherical panoramic image.&lt;ref&gt;{{cite web |title=Equirectangular Projection - PanoTools.org Wiki |url=https://wiki.panotools.org/Equirectangular_Projection |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

== Definition - Equirectangular Projection ==
* '''([[/SPH2EQUI/]])''' The forward projection transforms spherical coordinates into planar coordinates of the equirectangular projection. 
* '''([[/EQUI2SPH/]])''' The reverse projection transforms equirectangular coordinates from the plane back onto the sphere. The formulae presume a [[w:en:figure of the Earth|spherical model]] 

=== Spherical - Longitude and Latitude - SPH ===
* Longitude &lt;math&gt;\lambda \in [ -180^\circ , +180^\circ ] &lt;/math&gt;
* Latitude  &lt;math&gt;\phi \in [ -90^\circ , +90^\circ ]  &lt;/math&gt;

=== Visualization ===
A perspective view of the Earth showing how latitude (&lt;math&gt;\phi&lt;/math&gt;) and longitude (&lt;math&gt;\lambda&lt;/math&gt;) are defined on a spherical model. The graticule spacing is 10 degrees.
[[File:latitude and longitude graticule on a sphere.svg|center|350px|A perspective view of the Earth showing how latitude and longitude]]

==== Longitude - SPH ====
Longitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Greenich Meridian as the [[w:en:Prime Meridian|Prime Meridian]] and is ranging to &lt;math&gt;+180^o&lt;/math&gt; eastward and &lt;math&gt;-180^o&lt;/math&gt; westward. The Greek letter &lt;math&gt;\lambda&lt;/math&gt; (lambda)&lt;ref&gt;{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|title=Coordinate Conversion|website=colorado.edu|access-date=14 March 2018|archive-url=https://web.archive.org/web/20090929121405/http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|archive-date=29 September 2009|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;&quot;&lt;math&gt;\lambda&lt;/math&gt; = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign).&quot;&lt;br /&gt;John P. Snyder, ''[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 Map Projections, A Working Manual] {{Webarchive|url=https://web.archive.org/web/20100701103721/http://pubs.er.usgs.gov/usgspubs/pp/pp1395 |date=2010-07-01 }}'', [[w:en:USGS|USGS]] Professional Paper 1395, page ix&lt;/ref&gt; is used to denote the location of a place on Earth east or west of the Prime Meridian

==== Latitude - SPH ====
Latitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Equator and is ranging to &lt;math&gt;+90^o&lt;/math&gt; towards the North Pole and &lt;math&gt;-90^o&lt;/math&gt; towards the South Pole. The Greek letter &lt;math&gt;\phi&lt;/math&gt; or &lt;math&gt;\varphi&lt;/math&gt; (phi) denotes that angle.

==== Mnemonic - Greek Letter - Phi====
There are two different notations of the greek letter &lt;math&gt; \phi &lt;/math&gt; and &lt;math&gt; \varphi &lt;/math&gt;. In this learning resource the notation &lt;math&gt; \phi &lt;/math&gt; is used to indicate that it denotes the angle at circle that intersects with the North Pole and the South Pole.

=== Definition of Spherical Variables ===
The projections are [[w:en:Function_(mathematics)|mathematical function/mappings]]. For definition of these projections the following variables are defined:
*&lt;math&gt;\lambda&lt;/math&gt; is the [[w:en:longitude|longitude]] of the location to project;
* &lt;math&gt;\phi&lt;/math&gt; is the [[w:en:latitude|latitude]] of the location to project;
* &lt;math&gt;\phi_1&lt;/math&gt; are the standard parallels (north and south of the equator) where the scale of the projection is true;
* &lt;math&gt;\phi_0&lt;/math&gt; is the central parallel of the map (e.g. &lt;math&gt;\phi_0 = 0^\circ &lt;/math&gt; equator);
* &lt;math&gt;\lambda_0&lt;/math&gt; is the central meridian of the map;
* &lt;math&gt;R&lt;/math&gt; is the radius of the globe.
Longitude and latitude variables are defined here in terms of radians.

=== Definition of Equirectangular Planar Variables - EQUI ===
* &lt;math&gt;x_e&lt;/math&gt; is the horizontal coordinate of the projected location on the map;
* &lt;math&gt;y_e&lt;/math&gt; is the vertical coordinate of the projected location on the map;

=== Forward Projection - Spherical to Planar - SPH2EQUI ===
&lt;math&gt;\begin{align}
x &amp;= R \cdot (\lambda - \lambda_0) \cdot \cos (\phi_1)\\
y &amp;= R \cdot (\phi - \phi_0)
\end{align}&lt;/math&gt;

=== Special Case - Forward Projection ===
The {{lang|fr|plate carrée}} ([[w:en:French language|French]], for ''flat square''),&lt;ref&gt;{{Cite web |title=Plate Carrée - a simple example |last=Farkas |first=Gábor |work=O’Reilly Online Learning |date= |access-date=31 December 2022 |url= https://www.oreilly.com/library/view/practical-gis/9781787123328/Text/b21938a9-09f7-46fa-b905-58a0a4ed7d8f.xhtml}}&lt;/ref&gt; is the special case where &lt;math&gt;\varphi_1&lt;/math&gt; is zero. This projection maps ''x'' to be the value of the longitude and ''y'' to be the value of the latitude,&lt;ref&gt;{{cite book |url=https://books.google.co.uk/books?id=-FbVI-2tSuYC&amp;pg=PA119 |p=119 |title=Geographic Information Systems and Science |author1=Paul A. Longley |author2=Michael F. Goodchild |author3=David J. Maguire |author4=David W. Rhind |publisher=John Wiley &amp; Sons |year=2005}}&lt;/ref&gt; and therefore is sometimes called the latitude/longitude or lat/lon(g) projection. 

When the &lt;math&gt;\phi_1&lt;/math&gt; is not zero, such as [[w:en:Marinus of Tyre|Marinus]]'s &lt;math&gt;\phi_1=36&lt;/math&gt;,&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;7, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; or [[w:en:Royal Scottish Geographical Society|Ronald Miller]]'s &lt;math&gt;\phi_1=(37.5, 43.5, 50.5)&lt;/math&gt;,&lt;ref&gt;{{cite web |title=Equidistant Cylindrical (Plate Carrée) |url=https://proj.org/operations/projections/eqc.html |website=PROJ coordinate transformation software library |access-date=25 August 2020}}&lt;/ref&gt; the projection can portray particular latitudes of interest at true scale.

=== Remarks - Ellipsoidal Model ===
While a projection with equally spaced parallels is possible for an '''ellipsoidal model''', it would no longer be equidistant because the distance between parallels on an ellipsoid is not constant. More complex formulae can be used to create an equidistant map whose parallels reflect the true spacing.

===Reverse - Planar to Spherical - EQUI2SPH ===
&lt;math&gt;\begin{align}
\lambda &amp;= \frac{x} {R \cdot \cos (\phi_1)} + \lambda_0\\
\phi &amp;= \frac{y} {R} + \phi_0
\end{align}&lt;/math&gt;

=== Alternative names ===
In spherical panorama viewers, usually:

* &lt;math&gt;\lambda&lt;/math&gt; is called &quot;yaw&quot;;&lt;ref&gt;{{cite web |title=Yaw - PanoTools.org Wiki |url=https://wiki.panotools.org/Yaw |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;
* &lt;math&gt;\phi&lt;/math&gt; is called &quot;pitch&quot;;&lt;ref&gt;{{cite web |title=Pitch - PanoTools.org Wiki |url=https://wiki.panotools.org/Pitch |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

where both are defined in degrees.

== Learning Activities ==
The following learning activities address [[w:en:Projective geometry|Projective Geometry]] from a standard snapshot taken with your camera onto an area on the sphere according to the equirectangular projection. Keep in mind to distinguish between the following projective converters:
*  '''(SPH2EQUI / EQUI2SPH):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x_e,y_e)&lt;/math&gt; coordinates of an equirectangular image
*  '''(SPH2IMG / IMG2SPH):''' &lt;math&gt;(x_e,y_e)&lt;/math&gt; equirectangular coordinates &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.
*  '''(EQUI2IMG / IMG2EQUI):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.

=== Angle of View ===
For the learning activities it is important to understand the 
* '''(HAV)''' Horizontal Angle of View in landscape format and 
* '''(VAV)''' Vertical Angle of View in landscape format. 

=== Learning Task - Angle of View ===
The HAV and VAV differ from camera to camera. Explain why the HAV and VAV are relevant for the calculation of IMG2EQUI projection. Use the following figure to explain the requirements for projection.

=== Visualization of Angle of View ===
The following figure depicts besides the Horizontal (HAV) and Vertical Angle of View (VAV) also the Diagonal Angle of View (DAV).

[[File:Angle of view.svg|350px|center|Horizontal, vertical and diagonal angle of view]]

=== Remark - IMG2EQUI Projection ===
The current projection type of the learning activities is the (IMG2EQUI) projection between the &lt;math&gt;(x,y)&lt;/math&gt; coordinates of a standard image  and the distorted &lt;math&gt;(x_e,y_e)&lt;/math&gt;  coordinates of an equirectangular image and vice verca.

=== HAV and VAV as camera specifc properties ===
Due to the fact that different cameras have different angles of view (HAV and VAV) the visible area of the standard image (taken with your smartphone) might vary. The following animation shows different horizontal angles of view (HAV) e.g. for taking a snapshot vertically upwards towards the sky or blue ceiling.

[[File:Horizontal angle of view.gif|350px|center|dynamic visualization of the Horizontal Angle of View]]

=== IMG2EQUI Projection - Polar Regions ===
In this learning step we take pictures with a standard camera or smartphone and want to calculate the ''(Latitude,Longitude)'' coordinates of the sphere from ''(x,y)'' coordinates of standard image created with your smartphone or camera.
In the first learning step we consider the sphere at the polar regions. These regions are the most distorted areas in the equirectangular projections. 


=== IMG2SPH Projection - Take Snapshots for Learning Task ===
* Take your mobile phone and take two snapshots vertical down and vertical up from your floor and from the ceiling. Select an position in your room where the top and the floor has some visible elements (e.g. a lamp, wooden decorative elements,  ...). Alternatively you can create the two images outside with a cloudy sky and objects lying on the ground. 
* select a center of a circle and a radius that fits into both images (maximize the radius of the circle,
* in this module we will project these two images to the polar regions of the sphere within the rectangular coordinate system of the image.

==== Initial State - Image Sky ====
The following image shows the initial state taking a picture from the sky. The camera is located in the center of the red sphere and takes an image of the blue sky or the ceiling in a room. The ceiling is visualized as a blue plane. The learning task is to calculate the angles for the corresponding point on the sphere. 
 
[[File:Circle plane equirectangular.png|350px|Equirectangular Projection - Sky Image - Polar Projection]]

==== Side view with polar image ====
The side view can be used to calculate the angle. 

[[File:Equirectangular polar side view.gif|350px|Equirectangular projection side view - relevant angle is the red angle]]

==== Used Part of Snapshot with a Camera ====
The first image is snapshot with a camera taking a picture vertically upwards to the ceiling (into the sky). The red circle is the projected area from the image (sky/ceiling) onto a part of the equirectangular image visualized in the next section of the learning resource.

[[File:Equirectangular polar image.svg|350px|Equirectangular source image with circle to be projected]]

==== Equirectangular Projection of Circle - Rectangle ====
Explain, why the circle in the source image of the polar region (North Pole) is a rectangle after projection in the equirectangular image.

[[File:Equirectangular polar image projection.svg|350px|Equirectangular source image with circle to be projected]]

==== Learning Task - Function for Coordinate Transformation IMG2EQUI ====
We denote by &lt;math&gt;P_{IMG}(x,y)&lt;/math&gt; the color information of a pixel in the source image (IMG) and with &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt;  the color information of the pixel in the image in the destination format (EQUI). Define the function &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt; by calculating the corresponding &lt;math&gt;(x,y)&lt;/math&gt; coordinates in the IMG source image. This calculation is used to set pixels in the destination format EQUI by setting  
:&lt;math display=&quot;block&quot;&gt;P_{EQUI}(x_e,y_e):=P_{IMG}(x,y).&lt;/math&gt;
'''Remark:''' You need basic knowledge in [[trigonometry]] to perform this learning task.

==== Coordinate System of Graphics ====
[[File:Graphics coordinate system.svg|thumb|Coordinate system in Graphics]]
The coordinate system of an image has a different orientation in y-axis. This is shown in the diagram.
* &lt;math&gt;x_{max}&lt;/math&gt; is the maximal value of the on the x-axis of the image
* &lt;math&gt;y_{max}&lt;/math&gt; is the maximal value of the on the y-axis  the of the image
* The origin of the coordinate system is on the top left. 
Keep this in mind, when you use the coordinate system for your experiments with projections (see also equirectangular projection.

==== Rectangular to Sphere - Projection ====

[[File:Equirectangular ceiling floor.jpg|thumb|Equirectangular projection with marked ceiling and floor]]
Now we use a the standard rectangular image on the right as in input for the equirectangular projection on the sphere and we view the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image in panoramic preview on the sphere]. What can be observed, if you analyze the projected area of red [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg ceiling and a marked green rectangle on the floor].
* Drag the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image upwards to the ceiling] 
* and [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg downwards to floor].

==== Screenshots - Projection on Sphere of the standard Image ====
{| class=&quot;wikitable&quot;
|+ Screenshot of projection
|-
! Ceiling/Sky !! Floor
|-
| [[File:Equirectangular screenshot ceiling.png|320px|Screenshot ceiling/sky of standard image projected on a sphere with equirectangular projection]] || [[File:Equirectangular screenshot floor.png|320px|Screenshot floor of standard image projected on a sphere with equirectangular projection]]
|}

==== Learning Task - Projection of Sky in Graphics Coordinate System ====

[[File:Equirectangular coordinate system sky.svg|thumb|Equirectangular coordinate system sky / ceiling]]

Graphics have an own coordinate system to display points in pixel graphics or geometric objects like lines, polygons and circles in the coordinate system. The diagram shows the coordinate system of the image. Keep in mind that the coordinate of the y-axis has a different orientation than the y-axis in the standard 2D [[w:de:Cartesian coordinate system|Cartesian coordinate system]]. This information is relevant if you experiment with equirectangular projections in [[Projective Geometry Playground]].

Calculate the &lt;math&gt;y_{sky}&lt;/math&gt; for a specific angle of view of your camera.

==== Learning Task - Calculation ====
[[File:Horizontal angle of view.jpg|thumb|Fullscreen image of Door for HAV/VAV calc
[[File:Horizontal angle of view meaure distance.jpg|thumb|Measure distance from camera position to door ]]
ulation]]

Calculate the equirectangular projection for any &lt;math&gt;(x,y)&lt;/math&gt; point in the red circle of the source image IMG into the destination format EQUI with:
* '''LibreOffice - HAV/VAV:''' Take image of a door and calculate the horizontal and vertical angle of view (HAV and VAV)
* '''LibreOffice - Coordinates:''' Create a Spreadsheet document for the calculation of coordinates from a given coordinate in the equirectangular image the corresponding coordinates in the source image of your camera.
* [[Projective Geometry Playground|Javascript and HTML canvas]] - (see [[Projective Geometry Playground]]),
* [[w:en:GNU Octave|Octave]] with [https://gnu-octave.github.io/packages/image/ image-package]&lt;ref&gt; Carnë Draug,
Hartmut Gimpel, Avinoam Kalma (2022) Image Package Octave URL: https://gnu-octave.github.io/packages/image/ (March, 28th, 2024)&lt;/ref&gt;, or 
* Python with [https://github.com/python-pillow/Pillow Image Processing Library pillow]  by Jeffrey A. Clark   
Select an implementation of your choice. LibreOffice has the minimal requirements on programming skill but only coordinate transformation can be performed without a visual output.

=== Learning Task - Sky - North Pole ===
[[File:Sky image for equirectangular projection.jpg|thumb|Sky - equirectangular projection - north pole]]
A sky image can be used to project a circular area in the source image to the rectangular part at the top of the generated equirectangular image. 

Import the equirectangular projection in Aframe to preview the result.

==== Learning Task - Floor - South Pole ====
[[File:Floor sand for equirectangular.jpg|thumb|Floor Image - sand as demo input for equirectangular projection]]Transfer the lesson learned from north pole to the south pole and project the beach image as floor to the south pole. 
* What are differences and similarities between both projections?


== Final Result ==
[[File:Aldara parks.jpg|thumb|[https://niebert.github.io/HuginSample/Aldara_parks.html Equirectangular Image from Wikiversity used for Aframe 360 Degree Image] (see [[3D Modelling/Create 3D Models/Hugin|Hugin]])]]
{{PanoViewer|Frary Dining Hall 360-degree view.jpg|Dining Hall 360-degree view - with PanoViewer Template}}
* [https://niebert.github.io/HuginSample/ Final Results]  would can be previewed in Aframe (see [https://www.github.com/HuginSample HuginSample Files on Github]) or with the Wiki,
* [https://niebert.github.com/HuginSample/Aldara_parks.html Aldara Parks 360 Degree Image in Aframe] with an already uploaded equirectangular image in WikiMedia Commons by U.Bardins
* [https://panoviewer.toolforge.org/#Aldara_parks.jpg PanoViewer 360 degree view of Aldara Parks image]]

==See also==
* [[Projective Geometry Playground]]
* [[GeoGebra/Perspective Drawing on Mirror|Perspective Drawing on Mirror]]
* [[3D Modelling]]
* [[w:en:Cartography|Cartography]]
* [[w:en:Cassini projection|Cassini projection]]
* [[w:en:Gall–Peters projection|Gall–Peters projection]] with resolution regarding the use of rectangular world maps
* [[w:en:List of map projections|List of map projections]]
* [[w:en:Mercator projection|Mercator projection]]
* [[w:en:360 video projection|360 video projection]]
* [[3D_Modelling/Examples/Panorama_360|3D Modelling - 360 degree panorama]]
* [[Geometry]]
* [[w:en:Projective_geometry|Projective Geometry]]
* [[Portal:Mathematics]]
* [[w:en:GNU Octave|Octave]]
* [[Geogebra]]
* [[b:en:Javascript|Wikibook: Javascript]]
* [[Portal:Mathematics]]

==References==
{{Reflist}}

==External links==
* [https://visibleearth.nasa.gov/view.php?id=57730 Global MODIS based satellite map] The blue marble: land surface, ocean color, and sea ice.
* [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net.
* [http://wiki.panotools.org/Equirectangular Panoramic Equirectangular Projection], PanoTools wiki.
* [https://proj4.org/operations/projections/eqc.html Equidistant Cylindrical (Plate Carrée) in proj4]

== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/3D%20Modelling 3D Modelling]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Equirectangular%20projection https://en.wikiversity.org/wiki/Equirectangular%20projection]
--&gt;
* [https://en.wikiversity.org/wiki/Equirectangular%20projection This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
* Source: Wikiversity  https://en.wikiversity.org/wiki/Equirectangular%20projection
* see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal].
&lt;!-- * Next contents of the course are [[]] --&gt;;

=== Wikipedia2Wikiversiy===
This page was based on the following [https://en.wikipedia.org/wiki/Equirectangular%20projection wikipedia-source page]:
* [https://en.wikipedia.org/wiki/Equirectangular%20projection Equirectangular projection] https://en.wikipedia.org/wiki/Equirectangular%20projection
* Datum: 1/9/2023
* [https://niebert.github.io/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.io/Wikipedia2Wikiversity


[[Category:Map projections]]
[[Category:Equidistant projections]]
[[Category:Cylindrical projections]]
[[Category:Wiki2Reveal]]</text>
      <sha1>o6hf126nh3th2u73unvwux5ywc6fra8</sha1>
    </revision>
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      <text bytes="28424" xml:space="preserve">== Wiki2Reveal ==
This learning resource can be used as [[Wiki2Reveal]] slides in mathematics courses as introduction to [[w:en:Projective geometry|projective geometry]].

* Start '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=Geometry&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=Geometry Wiki2Reveal]''' [[File:Wiki2Reveal Logo.png|35px]]
  

== Introduction == 
In this learning resource the generation of an equirectangular projection is the objective. The learning resource is use-case driven, [https://niebert.github.io/HuginSample/ AFrame Example Durlach] is the first use-case of equirectangular projection. Look around by dragging the direction of view with the mouse with left mouse button pressed.
[[File:Hugin result in aframe.png|center|300px|Equirectangular image used in spheric image in a browser ]]
&lt;center&gt;
[https://niebert.github.io/HuginSample/ Preview of the equirectangular projection in AFrame] - Drag the preview of the equirectangular image with your mouse button pressed.
&lt;/center&gt;

=== Example of the equirectangular image ===  
Preview the [https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg equirectangular JPG-image] and explore the distortion of the image at the top and the bottom of the rectangular JPEG image. 

=== Navigation with multiple equirectangular images ===
The [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html AFrame Navigation example from the river Rhine in Germany] allows to jump from one location at the river rhine to another location at to the equirectangular preview.
 
[[File:river_rhine_spheric_preview.png|center|300px| - Use case of equirectangular projection  - River Rhine example with multiple locations]]

=== Equirectangular Projection for Maps ===
The following image shows an equirectangular projection of the world. The standard parallel is the equator (plate carrée projection)
[[File:Equirectangular projection SW.jpg|center|300px|upright=1.75|Equirectangular projection of the world; the standard parallel is the equator (plate carrée projection).]]

=== Distortion ===
Equirectangular projection with [[w:en:Tissot's indicatrix|Tissot's indicatrix]] of deformation and with the standard parallels lying on the equator. The deformation of a circle into an [[w:en:Ellipse|ellipse]] is visible on different location the world map. 
 
=== Deformation of a Circle - Distortion Indicator ===
The deformation of the circle is an indicator for the distortion of the image.
[[File:Plate Carrée with Tissot's Indicatrices of Distortion.svg|center|300px|upright=1.75|Equirectangular projection with Tissot's indicatrix of deformation and with the standard parallels lying on the equator]]

=== Areas of Interest ===
In the image above the distortion in the planar projection is
* minimal close to the equator and
* maximal at the south pole and north pole.
Rotating circle over the equator (e.g. intersecting with North and South Pole) can be used to have projections with minimal distortion in the area of interest.

=== North Pole and South Pole ===
The strongest distortion can be found at the north pole and south pole in the following true-colour satellite image of the earth. In the equirectangular projection the top horizontal line of pixels represent the single pixel for the north pole on the sphere model of the earth. Similar to that the  south pole as one pixel is stretched out bottom line of pixels in the equirectangular projection.

== Use-Case ==
From a set of standard images of the camera covering a full the from center point of view. In general you cover 360 degree circle with rectangular standard images and you take image with camera for the sky and the floor (preferred without seeing the tripod that might be used for the other images). These set of images are aggregate in one equirectangular image representing a [https://niebert.github.io/HuginSample/ full spheric panorama image], that be be viewed e.g. in [https://aframe.io/sky Aframe] or other panoramic OpenSource viewes, that support equirectangular images. 

=== EQUI2SPH Projection ===
The underlying type of projection is an equirectangular projection EQUI2SPH, that is used e.g in geographical context, where a sphere is projected to rectangular plane on the map. First of all we explore the use-case in [[3D Modelling]] about the spherical use of an spherical panorama image.

== Distortion ==
The projection creates especially at the &quot;North Pole&quot; and the &quot;South Pole&quot; the heaviest distortion in comparison to distances measured on the surface of the sphere. For panoramic views the distortion is just a matter of storage of the spheric pixel information in a rectangular format. On the image of a market place you will see that the panoramic viewes transfer the rectangular images into a natural view where you can look around and explore the location from different angles and with multiple equirectangular images from many locations (see [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html river Rhine example Cologne]).

== Origin of Terminology and History ==
The ''equirectangular projection'' (also called the ''equidistant cylindrical projection''), and which includes the special case of the ''plate carrée projection'' (also called the ''geographic projection'', ''lat/lon projection'', or ''plane chart''), is a simple [[w:en:map projection|map projection]] attributed to [[w:en:Marinus of Tyre|Marinus of Tyre]], who [[w:en:Ptolemy|Ptolemy]] claims invented the projection about AD 100.&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;5–8, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; The projection maps [[w:en:meridian (geography)|meridians]] to vertical straight lines of constant spacing (for meridional intervals of constant spacing), and [[w:en:circle of latitude|circles of latitude]] to horizontal straight lines of constant spacing (for constant intervals of [[w:en:circle of latitude|parallels]]). The projection is neither [[w:en:equal-area map|equal area]] nor [[w:en:conformal map projection|conformal]]. 

=== Implications of Distortion for Navigation ===
Because of the distortions introduced by this projection, it has little use in [[w:en:navigation|navigation]] or [[w:en:cadastral|cadastral]] mapping and finds its main use in [[w:en:thematic map|thematic mapping]]. 

=== Application in global Raster Datasets ===
In particular, the plate carrée has become a standard for global [[w:en:geographic information system|raster datasets]], such as [[w:en:Celestia|Celestia]], [[w:en:NASA World Wind|NASA World Wind]], the [[w:en:USGS|USGS]] [[w:en:Astrogeology Research Program|Astrogeology Research Program]], and [[w:en:Natural Earth|Natural Earth]], because of the particularly simple relationship between the position of an [[w:en:pixel|image pixel]] on the map and its corresponding geographic location on Earth or other spherical solar system bodies. 

=== Application in panoramic photography ===
In addition it is frequently used in panoramic photography to represent a spherical panoramic image.&lt;ref&gt;{{cite web |title=Equirectangular Projection - PanoTools.org Wiki |url=https://wiki.panotools.org/Equirectangular_Projection |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

== Definition - Equirectangular Projection ==
* '''([[/SPH2EQUI/]])''' The forward projection transforms spherical coordinates into planar coordinates of the equirectangular projection. 
* '''([[/EQUI2SPH/]])''' The reverse projection transforms equirectangular coordinates from the plane back onto the sphere. The formulae presume a [[w:en:figure of the Earth|spherical model]] 

=== Spherical - Longitude and Latitude - SPH ===
* Longitude &lt;math&gt;\lambda \in [ -180^\circ , +180^\circ ] &lt;/math&gt;
* Latitude  &lt;math&gt;\phi \in [ -90^\circ , +90^\circ ]  &lt;/math&gt;

=== Visualization ===
A perspective view of the Earth showing how latitude (&lt;math&gt;\phi&lt;/math&gt;) and longitude (&lt;math&gt;\lambda&lt;/math&gt;) are defined on a spherical model. The graticule spacing is 10 degrees.
[[File:latitude and longitude graticule on a sphere.svg|center|350px|A perspective view of the Earth showing how latitude and longitude]]

==== Longitude - SPH ====
Longitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Greenich Meridian as the [[w:en:Prime Meridian|Prime Meridian]] and is ranging to &lt;math&gt;+180^o&lt;/math&gt; eastward and &lt;math&gt;-180^o&lt;/math&gt; westward. The Greek letter &lt;math&gt;\lambda&lt;/math&gt; (lambda)&lt;ref&gt;{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|title=Coordinate Conversion|website=colorado.edu|access-date=14 March 2018|archive-url=https://web.archive.org/web/20090929121405/http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|archive-date=29 September 2009|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;&quot;&lt;math&gt;\lambda&lt;/math&gt; = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign).&quot;&lt;br /&gt;John P. Snyder, ''[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 Map Projections, A Working Manual] {{Webarchive|url=https://web.archive.org/web/20100701103721/http://pubs.er.usgs.gov/usgspubs/pp/pp1395 |date=2010-07-01 }}'', [[w:en:USGS|USGS]] Professional Paper 1395, page ix&lt;/ref&gt; is used to denote the location of a place on Earth east or west of the Prime Meridian

==== Latitude - SPH ====
Latitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Equator and is ranging to &lt;math&gt;+90^o&lt;/math&gt; towards the North Pole and &lt;math&gt;-90^o&lt;/math&gt; towards the South Pole. The Greek letter &lt;math&gt;\phi&lt;/math&gt; or &lt;math&gt;\varphi&lt;/math&gt; (phi) denotes that angle.

==== Mnemonic - Greek Letter - Phi====
There are two different notations of the greek letter &lt;math&gt; \phi &lt;/math&gt; and &lt;math&gt; \varphi &lt;/math&gt;. In this learning resource the notation &lt;math&gt; \phi &lt;/math&gt; is used to indicate that it denotes the angle at circle that intersects with the North Pole and the South Pole.

=== Definition of Spherical Variables ===
The projections are [[w:en:Function_(mathematics)|mathematical function/mappings]]. For definition of these projections the following variables are defined:
*&lt;math&gt;\lambda&lt;/math&gt; is the [[w:en:longitude|longitude]] of the location to project;
* &lt;math&gt;\phi&lt;/math&gt; is the [[w:en:latitude|latitude]] of the location to project;
* &lt;math&gt;\phi_1&lt;/math&gt; are the standard parallels (north and south of the equator) where the scale of the projection is true;
* &lt;math&gt;\phi_0&lt;/math&gt; is the central parallel of the map (e.g. &lt;math&gt;\phi_0 = 0^\circ &lt;/math&gt; equator);
* &lt;math&gt;\lambda_0&lt;/math&gt; is the central meridian of the map;
* &lt;math&gt;R&lt;/math&gt; is the radius of the globe.
Longitude and latitude variables are defined here in terms of radians.

=== Definition of Equirectangular Planar Variables - EQUI ===
* &lt;math&gt;x_e&lt;/math&gt; is the horizontal coordinate of the projected location on the map;
* &lt;math&gt;y_e&lt;/math&gt; is the vertical coordinate of the projected location on the map;

=== Forward Projection - Spherical to Planar - SPH2EQUI ===
&lt;math&gt;\begin{align}
x &amp;= R \cdot (\lambda - \lambda_0) \cdot \cos (\phi_1)\\
y &amp;= R \cdot (\phi - \phi_0)
\end{align}&lt;/math&gt;

=== Special Case - Forward Projection ===
The {{lang|fr|plate carrée}} ([[w:en:French language|French]], for ''flat square''),&lt;ref&gt;{{Cite web |title=Plate Carrée - a simple example |last=Farkas |first=Gábor |work=O’Reilly Online Learning |date= |access-date=31 December 2022 |url= https://www.oreilly.com/library/view/practical-gis/9781787123328/Text/b21938a9-09f7-46fa-b905-58a0a4ed7d8f.xhtml}}&lt;/ref&gt; is the special case where &lt;math&gt;\varphi_1&lt;/math&gt; is zero. This projection maps ''x'' to be the value of the longitude and ''y'' to be the value of the latitude,&lt;ref&gt;{{cite book |url=https://books.google.co.uk/books?id=-FbVI-2tSuYC&amp;pg=PA119 |p=119 |title=Geographic Information Systems and Science |author1=Paul A. Longley |author2=Michael F. Goodchild |author3=David J. Maguire |author4=David W. Rhind |publisher=John Wiley &amp; Sons |year=2005}}&lt;/ref&gt; and therefore is sometimes called the latitude/longitude or lat/lon(g) projection. 

When the &lt;math&gt;\phi_1&lt;/math&gt; is not zero, such as [[w:en:Marinus of Tyre|Marinus]]'s &lt;math&gt;\phi_1=36&lt;/math&gt;,&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;7, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; or [[w:en:Royal Scottish Geographical Society|Ronald Miller]]'s &lt;math&gt;\phi_1=(37.5, 43.5, 50.5)&lt;/math&gt;,&lt;ref&gt;{{cite web |title=Equidistant Cylindrical (Plate Carrée) |url=https://proj.org/operations/projections/eqc.html |website=PROJ coordinate transformation software library |access-date=25 August 2020}}&lt;/ref&gt; the projection can portray particular latitudes of interest at true scale.

=== Remarks - Ellipsoidal Model ===
While a projection with equally spaced parallels is possible for an '''ellipsoidal model''', it would no longer be equidistant because the distance between parallels on an ellipsoid is not constant. More complex formulae can be used to create an equidistant map whose parallels reflect the true spacing.

===Reverse - Planar to Spherical - EQUI2SPH ===
&lt;math&gt;\begin{align}
\lambda &amp;= \frac{x} {R \cdot \cos (\phi_1)} + \lambda_0\\
\phi &amp;= \frac{y} {R} + \phi_0
\end{align}&lt;/math&gt;

=== Alternative names ===
In spherical panorama viewers, usually:

* &lt;math&gt;\lambda&lt;/math&gt; is called &quot;yaw&quot;;&lt;ref&gt;{{cite web |title=Yaw - PanoTools.org Wiki |url=https://wiki.panotools.org/Yaw |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;
* &lt;math&gt;\phi&lt;/math&gt; is called &quot;pitch&quot;;&lt;ref&gt;{{cite web |title=Pitch - PanoTools.org Wiki |url=https://wiki.panotools.org/Pitch |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

where both are defined in degrees.

== Learning Activities ==
The following learning activities address [[w:en:Projective geometry|Projective Geometry]] from a standard snapshot taken with your camera onto an area on the sphere according to the equirectangular projection. Keep in mind to distinguish between the following projective converters:
*  '''(SPH2EQUI / EQUI2SPH):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x_e,y_e)&lt;/math&gt; coordinates of an equirectangular image
*  '''(SPH2IMG / IMG2SPH):''' &lt;math&gt;(x_e,y_e)&lt;/math&gt; equirectangular coordinates &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.
*  '''(EQUI2IMG / IMG2EQUI):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.

=== Angle of View ===
For the learning activities it is important to understand the 
* '''(HAV)''' Horizontal Angle of View in landscape format and 
* '''(VAV)''' Vertical Angle of View in landscape format. 

=== Learning Task - Angle of View ===
The HAV and VAV differ from camera to camera. Explain why the HAV and VAV are relevant for the calculation of IMG2EQUI projection. Use the following figure to explain the requirements for projection.

=== Visualization of Angle of View ===
The following figure depicts besides the Horizontal (HAV) and Vertical Angle of View (VAV) also the Diagonal Angle of View (DAV).

[[File:Angle of view.svg|350px|center|Horizontal, vertical and diagonal angle of view]]

=== Remark - IMG2EQUI Projection ===
The current projection type of the learning activities is the (IMG2EQUI) projection between the &lt;math&gt;(x,y)&lt;/math&gt; coordinates of a standard image  and the distorted &lt;math&gt;(x_e,y_e)&lt;/math&gt;  coordinates of an equirectangular image and vice verca.

=== HAV and VAV as camera specifc properties ===
Due to the fact that different cameras have different angles of view (HAV and VAV) the visible area of the standard image (taken with your smartphone) might vary. The following animation shows different horizontal angles of view (HAV) e.g. for taking a snapshot vertically upwards towards the sky or blue ceiling.

[[File:Horizontal angle of view.gif|350px|center|dynamic visualization of the Horizontal Angle of View]]

=== IMG2EQUI Projection - Polar Regions ===
In this learning step we take pictures with a standard camera or smartphone and want to calculate the ''(Latitude,Longitude)'' coordinates of the sphere from ''(x,y)'' coordinates of standard image created with your smartphone or camera.
In the first learning step we consider the sphere at the polar regions. These regions are the most distorted areas in the equirectangular projections. 


=== IMG2SPH Projection - Take Snapshots for Learning Task ===
* Take your mobile phone and take two snapshots vertical down and vertical up from your floor and from the ceiling. Select an position in your room where the top and the floor has some visible elements (e.g. a lamp, wooden decorative elements,  ...). Alternatively you can create the two images outside with a cloudy sky and objects lying on the ground. 
* select a center of a circle and a radius that fits into both images (maximize the radius of the circle,
* in this module we will project these two images to the polar regions of the sphere within the rectangular coordinate system of the image.

==== Initial State - Image Sky ====
The following image shows the initial state taking a picture from the sky. The camera is located in the center of the red sphere and takes an image of the blue sky or the ceiling in a room. The ceiling is visualized as a blue plane. The learning task is to calculate the angles for the corresponding point on the sphere. 
 
[[File:Circle plane equirectangular.png|350px|Equirectangular Projection - Sky Image - Polar Projection]]

==== Side view with polar image ====
The side view can be used to derive the calculation of the green angle with &lt;math&gt;\alpha:=\arctan(x)&lt;/math&gt;. The relevant angle for the equirectangular projection of the polar region is marked red. 

[[File:Equirectangular polar side view.gif|350px|Equirectangular projection side view - relevant angle is the red angle]]

==== Used Part of Snapshot with a Camera ====
The first image is snapshot with a camera taking a picture vertically upwards to the ceiling (into the sky). The red circle is the projected area from the image (sky/ceiling) onto a part of the equirectangular image visualized in the next section of the learning resource.

[[File:Equirectangular polar image.svg|350px|Equirectangular source image with circle to be projected]]

==== Equirectangular Projection of Circle - Rectangle ====
Explain, why the circle in the source image of the polar region (North Pole) is a rectangle after projection in the equirectangular image.

[[File:Equirectangular polar image projection.svg|350px|Equirectangular source image with circle to be projected]]

==== Learning Task - Function for Coordinate Transformation IMG2EQUI ====
We denote by &lt;math&gt;P_{IMG}(x,y)&lt;/math&gt; the color information of a pixel in the source image (IMG) and with &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt;  the color information of the pixel in the image in the destination format (EQUI). Define the function &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt; by calculating the corresponding &lt;math&gt;(x,y)&lt;/math&gt; coordinates in the IMG source image. This calculation is used to set pixels in the destination format EQUI by setting  
:&lt;math display=&quot;block&quot;&gt;P_{EQUI}(x_e,y_e):=P_{IMG}(x,y).&lt;/math&gt;
'''Remark:''' You need basic knowledge in [[trigonometry]] to perform this learning task.

==== Coordinate System of Graphics ====
[[File:Graphics coordinate system.svg|thumb|Coordinate system in Graphics]]
The coordinate system of an image has a different orientation in y-axis. This is shown in the diagram.
* &lt;math&gt;x_{max}&lt;/math&gt; is the maximal value of the on the x-axis of the image
* &lt;math&gt;y_{max}&lt;/math&gt; is the maximal value of the on the y-axis  the of the image
* The origin of the coordinate system is on the top left. 
Keep this in mind, when you use the coordinate system for your experiments with projections (see also equirectangular projection.

==== Rectangular to Sphere - Projection ====

[[File:Equirectangular ceiling floor.jpg|thumb|Equirectangular projection with marked ceiling and floor]]
Now we use a the standard rectangular image on the right as in input for the equirectangular projection on the sphere and we view the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image in panoramic preview on the sphere]. What can be observed, if you analyze the projected area of red [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg ceiling and a marked green rectangle on the floor].
* Drag the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image upwards to the ceiling] 
* and [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg downwards to floor].

==== Screenshots - Projection on Sphere of the standard Image ====
{| class=&quot;wikitable&quot;
|+ Screenshot of projection
|-
! Ceiling/Sky !! Floor
|-
| [[File:Equirectangular screenshot ceiling.png|320px|Screenshot ceiling/sky of standard image projected on a sphere with equirectangular projection]] || [[File:Equirectangular screenshot floor.png|320px|Screenshot floor of standard image projected on a sphere with equirectangular projection]]
|}

==== Learning Task - Projection of Sky in Graphics Coordinate System ====

[[File:Equirectangular coordinate system sky.svg|thumb|Equirectangular coordinate system sky / ceiling]]

Graphics have an own coordinate system to display points in pixel graphics or geometric objects like lines, polygons and circles in the coordinate system. The diagram shows the coordinate system of the image. Keep in mind that the coordinate of the y-axis has a different orientation than the y-axis in the standard 2D [[w:de:Cartesian coordinate system|Cartesian coordinate system]]. This information is relevant if you experiment with equirectangular projections in [[Projective Geometry Playground]].

Calculate the &lt;math&gt;y_{sky}&lt;/math&gt; for a specific angle of view of your camera.

==== Learning Task - Calculation ====
[[File:Horizontal angle of view.jpg|thumb|Fullscreen image of Door for HAV/VAV calc
[[File:Horizontal angle of view meaure distance.jpg|thumb|Measure distance from camera position to door ]]
ulation]]

Calculate the equirectangular projection for any &lt;math&gt;(x,y)&lt;/math&gt; point in the red circle of the source image IMG into the destination format EQUI with:
* '''LibreOffice - HAV/VAV:''' Take image of a door and calculate the horizontal and vertical angle of view (HAV and VAV)
* '''LibreOffice - Coordinates:''' Create a Spreadsheet document for the calculation of coordinates from a given coordinate in the equirectangular image the corresponding coordinates in the source image of your camera.
* [[Projective Geometry Playground|Javascript and HTML canvas]] - (see [[Projective Geometry Playground]]),
* [[w:en:GNU Octave|Octave]] with [https://gnu-octave.github.io/packages/image/ image-package]&lt;ref&gt; Carnë Draug,
Hartmut Gimpel, Avinoam Kalma (2022) Image Package Octave URL: https://gnu-octave.github.io/packages/image/ (March, 28th, 2024)&lt;/ref&gt;, or 
* Python with [https://github.com/python-pillow/Pillow Image Processing Library pillow]  by Jeffrey A. Clark   
Select an implementation of your choice. LibreOffice has the minimal requirements on programming skill but only coordinate transformation can be performed without a visual output.

=== Learning Task - Sky - North Pole ===
[[File:Sky image for equirectangular projection.jpg|thumb|Sky - equirectangular projection - north pole]]
A sky image can be used to project a circular area in the source image to the rectangular part at the top of the generated equirectangular image. 

Import the equirectangular projection in Aframe to preview the result.

==== Learning Task - Floor - South Pole ====
[[File:Floor sand for equirectangular.jpg|thumb|Floor Image - sand as demo input for equirectangular projection]]Transfer the lesson learned from north pole to the south pole and project the beach image as floor to the south pole. 
* What are differences and similarities between both projections?


== Final Result ==
[[File:Aldara parks.jpg|thumb|[https://niebert.github.io/HuginSample/Aldara_parks.html Equirectangular Image from Wikiversity used for Aframe 360 Degree Image] (see [[3D Modelling/Create 3D Models/Hugin|Hugin]])]]
{{PanoViewer|Frary Dining Hall 360-degree view.jpg|Dining Hall 360-degree view - with PanoViewer Template}}
* [https://niebert.github.io/HuginSample/ Final Results]  would can be previewed in Aframe (see [https://www.github.com/HuginSample HuginSample Files on Github]) or with the Wiki,
* [https://niebert.github.com/HuginSample/Aldara_parks.html Aldara Parks 360 Degree Image in Aframe] with an already uploaded equirectangular image in WikiMedia Commons by U.Bardins
* [https://panoviewer.toolforge.org/#Aldara_parks.jpg PanoViewer 360 degree view of Aldara Parks image]]

==See also==
* [[Projective Geometry Playground]]
* [[GeoGebra/Perspective Drawing on Mirror|Perspective Drawing on Mirror]]
* [[3D Modelling]]
* [[w:en:Cartography|Cartography]]
* [[w:en:Cassini projection|Cassini projection]]
* [[w:en:Gall–Peters projection|Gall–Peters projection]] with resolution regarding the use of rectangular world maps
* [[w:en:List of map projections|List of map projections]]
* [[w:en:Mercator projection|Mercator projection]]
* [[w:en:360 video projection|360 video projection]]
* [[3D_Modelling/Examples/Panorama_360|3D Modelling - 360 degree panorama]]
* [[Geometry]]
* [[w:en:Projective_geometry|Projective Geometry]]
* [[Portal:Mathematics]]
* [[w:en:GNU Octave|Octave]]
* [[Geogebra]]
* [[b:en:Javascript|Wikibook: Javascript]]
* [[Portal:Mathematics]]

==References==
{{Reflist}}

==External links==
* [https://visibleearth.nasa.gov/view.php?id=57730 Global MODIS based satellite map] The blue marble: land surface, ocean color, and sea ice.
* [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net.
* [http://wiki.panotools.org/Equirectangular Panoramic Equirectangular Projection], PanoTools wiki.
* [https://proj4.org/operations/projections/eqc.html Equidistant Cylindrical (Plate Carrée) in proj4]

== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/3D%20Modelling 3D Modelling]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Equirectangular%20projection https://en.wikiversity.org/wiki/Equirectangular%20projection]
--&gt;
* [https://en.wikiversity.org/wiki/Equirectangular%20projection This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
* Source: Wikiversity  https://en.wikiversity.org/wiki/Equirectangular%20projection
* see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal].
&lt;!-- * Next contents of the course are [[]] --&gt;;

=== Wikipedia2Wikiversiy===
This page was based on the following [https://en.wikipedia.org/wiki/Equirectangular%20projection wikipedia-source page]:
* [https://en.wikipedia.org/wiki/Equirectangular%20projection Equirectangular projection] https://en.wikipedia.org/wiki/Equirectangular%20projection
* Datum: 1/9/2023
* [https://niebert.github.io/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.io/Wikipedia2Wikiversity


[[Category:Map projections]]
[[Category:Equidistant projections]]
[[Category:Cylindrical projections]]
[[Category:Wiki2Reveal]]</text>
      <sha1>kt65myxhn7znc8w604czeu7ietdd1dg</sha1>
    </revision>
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      <text bytes="26725" xml:space="preserve">== Wiki2Reveal ==
This learning resource can be used as [[Wiki2Reveal]] slides in mathematics courses as introduction to [[w:en:Projective geometry|projective geometry]].

* Start '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=Geometry&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=Geometry Wiki2Reveal]''' [[File:Wiki2Reveal Logo.png|35px]]
  

== Introduction == 
In this learning resource the generation of an equirectangular projection is the objective. The learning resource is use-case driven, [https://niebert.github.io/HuginSample/ AFrame Example Durlach] is the first use-case of equirectangular projection. Look around by dragging the direction of view with the mouse with left mouse button pressed.
[[File:Hugin result in aframe.png|center|300px|Equirectangular image used in spheric image in a browser ]]
&lt;center&gt;
[https://niebert.github.io/HuginSample/ Preview of the equirectangular projection in AFrame] - Drag the preview of the equirectangular image with your mouse button pressed.
&lt;/center&gt;

=== Example of the equirectangular image ===  
Preview the [https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg equirectangular JPG-image] and explore the distortion of the image at the top and the bottom of the rectangular JPEG image. 

=== Navigation with multiple equirectangular images ===
The [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html AFrame Navigation example from the river Rhine in Germany] allows to jump from one location at the river rhine to another location at to the equirectangular preview.
 
[[File:river_rhine_spheric_preview.png|center|300px| - Use case of equirectangular projection  - River Rhine example with multiple locations]]

== Subtopics ==
* [[/Maps and Distortion/]]

== Use-Case ==
From a set of standard images of the camera covering a full the from center point of view. In general you cover 360 degree circle with rectangular standard images and you take image with camera for the sky and the floor (preferred without seeing the tripod that might be used for the other images). These set of images are aggregate in one equirectangular image representing a [https://niebert.github.io/HuginSample/ full spheric panorama image], that be be viewed e.g. in [https://aframe.io/sky Aframe] or other panoramic OpenSource viewes, that support equirectangular images. 

=== EQUI2SPH Projection ===
The underlying type of projection is an equirectangular projection EQUI2SPH, that is used e.g in geographical context, where a sphere is projected to rectangular plane on the map. First of all we explore the use-case in [[3D Modelling]] about the spherical use of an spherical panorama image.

== Distortion ==
The projection creates especially at the &quot;North Pole&quot; and the &quot;South Pole&quot; the heaviest distortion in comparison to distances measured on the surface of the sphere. For panoramic views the distortion is just a matter of storage of the spheric pixel information in a rectangular format. On the image of a market place you will see that the panoramic viewes transfer the rectangular images into a natural view where you can look around and explore the location from different angles and with multiple equirectangular images from many locations (see [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html river Rhine example Cologne]).

== Origin of Terminology and History ==
The ''equirectangular projection'' (also called the ''equidistant cylindrical projection''), and which includes the special case of the ''plate carrée projection'' (also called the ''geographic projection'', ''lat/lon projection'', or ''plane chart''), is a simple [[w:en:map projection|map projection]] attributed to [[w:en:Marinus of Tyre|Marinus of Tyre]], who [[w:en:Ptolemy|Ptolemy]] claims invented the projection about AD 100.&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;5–8, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; The projection maps [[w:en:meridian (geography)|meridians]] to vertical straight lines of constant spacing (for meridional intervals of constant spacing), and [[w:en:circle of latitude|circles of latitude]] to horizontal straight lines of constant spacing (for constant intervals of [[w:en:circle of latitude|parallels]]). The projection is neither [[w:en:equal-area map|equal area]] nor [[w:en:conformal map projection|conformal]]. 

=== Implications of Distortion for Navigation ===
Because of the distortions introduced by this projection, it has little use in [[w:en:navigation|navigation]] or [[w:en:cadastral|cadastral]] mapping and finds its main use in [[w:en:thematic map|thematic mapping]]. 

=== Application in global Raster Datasets ===
In particular, the plate carrée has become a standard for global [[w:en:geographic information system|raster datasets]], such as [[w:en:Celestia|Celestia]], [[w:en:NASA World Wind|NASA World Wind]], the [[w:en:USGS|USGS]] [[w:en:Astrogeology Research Program|Astrogeology Research Program]], and [[w:en:Natural Earth|Natural Earth]], because of the particularly simple relationship between the position of an [[w:en:pixel|image pixel]] on the map and its corresponding geographic location on Earth or other spherical solar system bodies. 

=== Application in panoramic photography ===
In addition it is frequently used in panoramic photography to represent a spherical panoramic image.&lt;ref&gt;{{cite web |title=Equirectangular Projection - PanoTools.org Wiki |url=https://wiki.panotools.org/Equirectangular_Projection |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

== Definition - Equirectangular Projection ==
* '''([[/SPH2EQUI/]])''' The forward projection transforms spherical coordinates into planar coordinates of the equirectangular projection. 
* '''([[/EQUI2SPH/]])''' The reverse projection transforms equirectangular coordinates from the plane back onto the sphere. The formulae presume a [[w:en:figure of the Earth|spherical model]] 

=== Spherical - Longitude and Latitude - SPH ===
* Longitude &lt;math&gt;\lambda \in [ -180^\circ , +180^\circ ] &lt;/math&gt;
* Latitude  &lt;math&gt;\phi \in [ -90^\circ , +90^\circ ]  &lt;/math&gt;

=== Visualization ===
A perspective view of the Earth showing how latitude (&lt;math&gt;\phi&lt;/math&gt;) and longitude (&lt;math&gt;\lambda&lt;/math&gt;) are defined on a spherical model. The graticule spacing is 10 degrees.
[[File:latitude and longitude graticule on a sphere.svg|center|350px|A perspective view of the Earth showing how latitude and longitude]]

==== Longitude - SPH ====
Longitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Greenich Meridian as the [[w:en:Prime Meridian|Prime Meridian]] and is ranging to &lt;math&gt;+180^o&lt;/math&gt; eastward and &lt;math&gt;-180^o&lt;/math&gt; westward. The Greek letter &lt;math&gt;\lambda&lt;/math&gt; (lambda)&lt;ref&gt;{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|title=Coordinate Conversion|website=colorado.edu|access-date=14 March 2018|archive-url=https://web.archive.org/web/20090929121405/http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|archive-date=29 September 2009|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;&quot;&lt;math&gt;\lambda&lt;/math&gt; = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign).&quot;&lt;br /&gt;John P. Snyder, ''[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 Map Projections, A Working Manual] {{Webarchive|url=https://web.archive.org/web/20100701103721/http://pubs.er.usgs.gov/usgspubs/pp/pp1395 |date=2010-07-01 }}'', [[w:en:USGS|USGS]] Professional Paper 1395, page ix&lt;/ref&gt; is used to denote the location of a place on Earth east or west of the Prime Meridian

==== Latitude - SPH ====
Latitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Equator and is ranging to &lt;math&gt;+90^o&lt;/math&gt; towards the North Pole and &lt;math&gt;-90^o&lt;/math&gt; towards the South Pole. The Greek letter &lt;math&gt;\phi&lt;/math&gt; or &lt;math&gt;\varphi&lt;/math&gt; (phi) denotes that angle.

==== Mnemonic - Greek Letter - Phi====
There are two different notations of the greek letter &lt;math&gt; \phi &lt;/math&gt; and &lt;math&gt; \varphi &lt;/math&gt;. In this learning resource the notation &lt;math&gt; \phi &lt;/math&gt; is used to indicate that it denotes the angle at circle that intersects with the North Pole and the South Pole.

=== Definition of Spherical Variables ===
The projections are [[w:en:Function_(mathematics)|mathematical function/mappings]]. For definition of these projections the following variables are defined:
*&lt;math&gt;\lambda&lt;/math&gt; is the [[w:en:longitude|longitude]] of the location to project;
* &lt;math&gt;\phi&lt;/math&gt; is the [[w:en:latitude|latitude]] of the location to project;
* &lt;math&gt;\phi_1&lt;/math&gt; are the standard parallels (north and south of the equator) where the scale of the projection is true;
* &lt;math&gt;\phi_0&lt;/math&gt; is the central parallel of the map (e.g. &lt;math&gt;\phi_0 = 0^\circ &lt;/math&gt; equator);
* &lt;math&gt;\lambda_0&lt;/math&gt; is the central meridian of the map;
* &lt;math&gt;R&lt;/math&gt; is the radius of the globe.
Longitude and latitude variables are defined here in terms of radians.

=== Definition of Equirectangular Planar Variables - EQUI ===
* &lt;math&gt;x_e&lt;/math&gt; is the horizontal coordinate of the projected location on the map;
* &lt;math&gt;y_e&lt;/math&gt; is the vertical coordinate of the projected location on the map;

=== Forward Projection - Spherical to Planar - SPH2EQUI ===
&lt;math&gt;\begin{align}
x &amp;= R \cdot (\lambda - \lambda_0) \cdot \cos (\phi_1)\\
y &amp;= R \cdot (\phi - \phi_0)
\end{align}&lt;/math&gt;

=== Special Case - Forward Projection ===
The {{lang|fr|plate carrée}} ([[w:en:French language|French]], for ''flat square''),&lt;ref&gt;{{Cite web |title=Plate Carrée - a simple example |last=Farkas |first=Gábor |work=O’Reilly Online Learning |date= |access-date=31 December 2022 |url= https://www.oreilly.com/library/view/practical-gis/9781787123328/Text/b21938a9-09f7-46fa-b905-58a0a4ed7d8f.xhtml}}&lt;/ref&gt; is the special case where &lt;math&gt;\varphi_1&lt;/math&gt; is zero. This projection maps ''x'' to be the value of the longitude and ''y'' to be the value of the latitude,&lt;ref&gt;{{cite book |url=https://books.google.co.uk/books?id=-FbVI-2tSuYC&amp;pg=PA119 |p=119 |title=Geographic Information Systems and Science |author1=Paul A. Longley |author2=Michael F. Goodchild |author3=David J. Maguire |author4=David W. Rhind |publisher=John Wiley &amp; Sons |year=2005}}&lt;/ref&gt; and therefore is sometimes called the latitude/longitude or lat/lon(g) projection. 

When the &lt;math&gt;\phi_1&lt;/math&gt; is not zero, such as [[w:en:Marinus of Tyre|Marinus]]'s &lt;math&gt;\phi_1=36&lt;/math&gt;,&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;7, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; or [[w:en:Royal Scottish Geographical Society|Ronald Miller]]'s &lt;math&gt;\phi_1=(37.5, 43.5, 50.5)&lt;/math&gt;,&lt;ref&gt;{{cite web |title=Equidistant Cylindrical (Plate Carrée) |url=https://proj.org/operations/projections/eqc.html |website=PROJ coordinate transformation software library |access-date=25 August 2020}}&lt;/ref&gt; the projection can portray particular latitudes of interest at true scale.

=== Remarks - Ellipsoidal Model ===
While a projection with equally spaced parallels is possible for an '''ellipsoidal model''', it would no longer be equidistant because the distance between parallels on an ellipsoid is not constant. More complex formulae can be used to create an equidistant map whose parallels reflect the true spacing.

===Reverse - Planar to Spherical - EQUI2SPH ===
&lt;math&gt;\begin{align}
\lambda &amp;= \frac{x} {R \cdot \cos (\phi_1)} + \lambda_0\\
\phi &amp;= \frac{y} {R} + \phi_0
\end{align}&lt;/math&gt;

=== Alternative names ===
In spherical panorama viewers, usually:

* &lt;math&gt;\lambda&lt;/math&gt; is called &quot;yaw&quot;;&lt;ref&gt;{{cite web |title=Yaw - PanoTools.org Wiki |url=https://wiki.panotools.org/Yaw |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;
* &lt;math&gt;\phi&lt;/math&gt; is called &quot;pitch&quot;;&lt;ref&gt;{{cite web |title=Pitch - PanoTools.org Wiki |url=https://wiki.panotools.org/Pitch |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

where both are defined in degrees.

== Learning Activities ==
The following learning activities address [[w:en:Projective geometry|Projective Geometry]] from a standard snapshot taken with your camera onto an area on the sphere according to the equirectangular projection. Keep in mind to distinguish between the following projective converters:
*  '''(SPH2EQUI / EQUI2SPH):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x_e,y_e)&lt;/math&gt; coordinates of an equirectangular image
*  '''(SPH2IMG / IMG2SPH):''' &lt;math&gt;(x_e,y_e)&lt;/math&gt; equirectangular coordinates &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.
*  '''(EQUI2IMG / IMG2EQUI):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.

=== Angle of View ===
For the learning activities it is important to understand the 
* '''(HAV)''' Horizontal Angle of View in landscape format and 
* '''(VAV)''' Vertical Angle of View in landscape format. 

=== Learning Task - Angle of View ===
The HAV and VAV differ from camera to camera. Explain why the HAV and VAV are relevant for the calculation of IMG2EQUI projection. Use the following figure to explain the requirements for projection.

=== Visualization of Angle of View ===
The following figure depicts besides the Horizontal (HAV) and Vertical Angle of View (VAV) also the Diagonal Angle of View (DAV).

[[File:Angle of view.svg|350px|center|Horizontal, vertical and diagonal angle of view]]

=== Remark - IMG2EQUI Projection ===
The current projection type of the learning activities is the (IMG2EQUI) projection between the &lt;math&gt;(x,y)&lt;/math&gt; coordinates of a standard image  and the distorted &lt;math&gt;(x_e,y_e)&lt;/math&gt;  coordinates of an equirectangular image and vice verca.

=== HAV and VAV as camera specifc properties ===
Due to the fact that different cameras have different angles of view (HAV and VAV) the visible area of the standard image (taken with your smartphone) might vary. The following animation shows different horizontal angles of view (HAV) e.g. for taking a snapshot vertically upwards towards the sky or blue ceiling.

[[File:Horizontal angle of view.gif|350px|center|dynamic visualization of the Horizontal Angle of View]]

=== IMG2EQUI Projection - Polar Regions ===
In this learning step we take pictures with a standard camera or smartphone and want to calculate the ''(Latitude,Longitude)'' coordinates of the sphere from ''(x,y)'' coordinates of standard image created with your smartphone or camera.
In the first learning step we consider the sphere at the polar regions. These regions are the most distorted areas in the equirectangular projections. 


=== IMG2SPH Projection - Take Snapshots for Learning Task ===
* Take your mobile phone and take two snapshots vertical down and vertical up from your floor and from the ceiling. Select an position in your room where the top and the floor has some visible elements (e.g. a lamp, wooden decorative elements,  ...). Alternatively you can create the two images outside with a cloudy sky and objects lying on the ground. 
* select a center of a circle and a radius that fits into both images (maximize the radius of the circle,
* in this module we will project these two images to the polar regions of the sphere within the rectangular coordinate system of the image.

==== Initial State - Image Sky ====
The following image shows the initial state taking a picture from the sky. The camera is located in the center of the red sphere and takes an image of the blue sky or the ceiling in a room. The ceiling is visualized as a blue plane. The learning task is to calculate the angles for the corresponding point on the sphere. 
 
[[File:Circle plane equirectangular.png|350px|Equirectangular Projection - Sky Image - Polar Projection]]

==== Side view with polar image ====
The side view can be used to derive the calculation of the green angle with &lt;math&gt;\alpha:=\arctan(x)&lt;/math&gt;. The relevant angle for the equirectangular projection of the polar region is marked red. 

[[File:Equirectangular polar side view.gif|350px|Equirectangular projection side view - relevant angle is the red angle]]

==== Used Part of Snapshot with a Camera ====
The first image is snapshot with a camera taking a picture vertically upwards to the ceiling (into the sky). The red circle is the projected area from the image (sky/ceiling) onto a part of the equirectangular image visualized in the next section of the learning resource.

[[File:Equirectangular polar image.svg|350px|Equirectangular source image with circle to be projected]]

==== Equirectangular Projection of Circle - Rectangle ====
Explain, why the circle in the source image of the polar region (North Pole) is a rectangle after projection in the equirectangular image.

[[File:Equirectangular polar image projection.svg|350px|Equirectangular source image with circle to be projected]]

==== Learning Task - Function for Coordinate Transformation IMG2EQUI ====
We denote by &lt;math&gt;P_{IMG}(x,y)&lt;/math&gt; the color information of a pixel in the source image (IMG) and with &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt;  the color information of the pixel in the image in the destination format (EQUI). Define the function &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt; by calculating the corresponding &lt;math&gt;(x,y)&lt;/math&gt; coordinates in the IMG source image. This calculation is used to set pixels in the destination format EQUI by setting  
:&lt;math display=&quot;block&quot;&gt;P_{EQUI}(x_e,y_e):=P_{IMG}(x,y).&lt;/math&gt;
'''Remark:''' You need basic knowledge in [[trigonometry]] to perform this learning task.

==== Coordinate System of Graphics ====
[[File:Graphics coordinate system.svg|thumb|Coordinate system in Graphics]]
The coordinate system of an image has a different orientation in y-axis. This is shown in the diagram.
* &lt;math&gt;x_{max}&lt;/math&gt; is the maximal value of the on the x-axis of the image
* &lt;math&gt;y_{max}&lt;/math&gt; is the maximal value of the on the y-axis  the of the image
* The origin of the coordinate system is on the top left. 
Keep this in mind, when you use the coordinate system for your experiments with projections (see also equirectangular projection.

==== Rectangular to Sphere - Projection ====

[[File:Equirectangular ceiling floor.jpg|thumb|Equirectangular projection with marked ceiling and floor]]
Now we use a the standard rectangular image on the right as in input for the equirectangular projection on the sphere and we view the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image in panoramic preview on the sphere]. What can be observed, if you analyze the projected area of red [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg ceiling and a marked green rectangle on the floor].
* Drag the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image upwards to the ceiling] 
* and [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg downwards to floor].

==== Screenshots - Projection on Sphere of the standard Image ====
{| class=&quot;wikitable&quot;
|+ Screenshot of projection
|-
! Ceiling/Sky !! Floor
|-
| [[File:Equirectangular screenshot ceiling.png|320px|Screenshot ceiling/sky of standard image projected on a sphere with equirectangular projection]] || [[File:Equirectangular screenshot floor.png|320px|Screenshot floor of standard image projected on a sphere with equirectangular projection]]
|}

==== Learning Task - Projection of Sky in Graphics Coordinate System ====

[[File:Equirectangular coordinate system sky.svg|thumb|Equirectangular coordinate system sky / ceiling]]

Graphics have an own coordinate system to display points in pixel graphics or geometric objects like lines, polygons and circles in the coordinate system. The diagram shows the coordinate system of the image. Keep in mind that the coordinate of the y-axis has a different orientation than the y-axis in the standard 2D [[w:de:Cartesian coordinate system|Cartesian coordinate system]]. This information is relevant if you experiment with equirectangular projections in [[Projective Geometry Playground]].

Calculate the &lt;math&gt;y_{sky}&lt;/math&gt; for a specific angle of view of your camera.

==== Learning Task - Calculation ====
[[File:Horizontal angle of view.jpg|thumb|Fullscreen image of Door for HAV/VAV calc
[[File:Horizontal angle of view meaure distance.jpg|thumb|Measure distance from camera position to door ]]
ulation]]

Calculate the equirectangular projection for any &lt;math&gt;(x,y)&lt;/math&gt; point in the red circle of the source image IMG into the destination format EQUI with:
* '''LibreOffice - HAV/VAV:''' Take image of a door and calculate the horizontal and vertical angle of view (HAV and VAV)
* '''LibreOffice - Coordinates:''' Create a Spreadsheet document for the calculation of coordinates from a given coordinate in the equirectangular image the corresponding coordinates in the source image of your camera.
* [[Projective Geometry Playground|Javascript and HTML canvas]] - (see [[Projective Geometry Playground]]),
* [[w:en:GNU Octave|Octave]] with [https://gnu-octave.github.io/packages/image/ image-package]&lt;ref&gt; Carnë Draug,
Hartmut Gimpel, Avinoam Kalma (2022) Image Package Octave URL: https://gnu-octave.github.io/packages/image/ (March, 28th, 2024)&lt;/ref&gt;, or 
* Python with [https://github.com/python-pillow/Pillow Image Processing Library pillow]  by Jeffrey A. Clark   
Select an implementation of your choice. LibreOffice has the minimal requirements on programming skill but only coordinate transformation can be performed without a visual output.

=== Learning Task - Sky - North Pole ===
[[File:Sky image for equirectangular projection.jpg|thumb|Sky - equirectangular projection - north pole]]
A sky image can be used to project a circular area in the source image to the rectangular part at the top of the generated equirectangular image. 

Import the equirectangular projection in Aframe to preview the result.

==== Learning Task - Floor - South Pole ====
[[File:Floor sand for equirectangular.jpg|thumb|Floor Image - sand as demo input for equirectangular projection]]Transfer the lesson learned from north pole to the south pole and project the beach image as floor to the south pole. 
* What are differences and similarities between both projections?


== Final Result ==
[[File:Aldara parks.jpg|thumb|[https://niebert.github.io/HuginSample/Aldara_parks.html Equirectangular Image from Wikiversity used for Aframe 360 Degree Image] (see [[3D Modelling/Create 3D Models/Hugin|Hugin]])]]
{{PanoViewer|Frary Dining Hall 360-degree view.jpg|Dining Hall 360-degree view - with PanoViewer Template}}
* [https://niebert.github.io/HuginSample/ Final Results]  would can be previewed in Aframe (see [https://www.github.com/HuginSample HuginSample Files on Github]) or with the Wiki,
* [https://niebert.github.com/HuginSample/Aldara_parks.html Aldara Parks 360 Degree Image in Aframe] with an already uploaded equirectangular image in WikiMedia Commons by U.Bardins
* [https://panoviewer.toolforge.org/#Aldara_parks.jpg PanoViewer 360 degree view of Aldara Parks image]]

==See also==
* [[Projective Geometry Playground]]
* [[GeoGebra/Perspective Drawing on Mirror|Perspective Drawing on Mirror]]
* [[3D Modelling]]
* [[w:en:Cartography|Cartography]]
* [[w:en:Cassini projection|Cassini projection]]
* [[w:en:Gall–Peters projection|Gall–Peters projection]] with resolution regarding the use of rectangular world maps
* [[w:en:List of map projections|List of map projections]]
* [[w:en:Mercator projection|Mercator projection]]
* [[w:en:360 video projection|360 video projection]]
* [[3D_Modelling/Examples/Panorama_360|3D Modelling - 360 degree panorama]]
* [[Geometry]]
* [[w:en:Projective_geometry|Projective Geometry]]
* [[Portal:Mathematics]]
* [[w:en:GNU Octave|Octave]]
* [[Geogebra]]
* [[b:en:Javascript|Wikibook: Javascript]]
* [[Portal:Mathematics]]

==References==
{{Reflist}}

==External links==
* [https://visibleearth.nasa.gov/view.php?id=57730 Global MODIS based satellite map] The blue marble: land surface, ocean color, and sea ice.
* [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net.
* [http://wiki.panotools.org/Equirectangular Panoramic Equirectangular Projection], PanoTools wiki.
* [https://proj4.org/operations/projections/eqc.html Equidistant Cylindrical (Plate Carrée) in proj4]

== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/3D%20Modelling 3D Modelling]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Equirectangular%20projection https://en.wikiversity.org/wiki/Equirectangular%20projection]
--&gt;
* [https://en.wikiversity.org/wiki/Equirectangular%20projection This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
* Source: Wikiversity  https://en.wikiversity.org/wiki/Equirectangular%20projection
* see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal].
&lt;!-- * Next contents of the course are [[]] --&gt;;

=== Wikipedia2Wikiversiy===
This page was based on the following [https://en.wikipedia.org/wiki/Equirectangular%20projection wikipedia-source page]:
* [https://en.wikipedia.org/wiki/Equirectangular%20projection Equirectangular projection] https://en.wikipedia.org/wiki/Equirectangular%20projection
* Datum: 1/9/2023
* [https://niebert.github.io/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.io/Wikipedia2Wikiversity


[[Category:Map projections]]
[[Category:Equidistant projections]]
[[Category:Cylindrical projections]]
[[Category:Wiki2Reveal]]</text>
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      <text bytes="26057" xml:space="preserve">== Wiki2Reveal ==
This learning resource can be used as [[Wiki2Reveal]] slides in mathematics courses as introduction to [[w:en:Projective geometry|projective geometry]].

* Start '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=Geometry&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=Geometry Wiki2Reveal]''' [[File:Wiki2Reveal Logo.png|35px]]
  

== Introduction == 
In this learning resource the generation of an equirectangular projection is the objective. The learning resource is use-case driven, [https://niebert.github.io/HuginSample/ AFrame Example Durlach] is the first use-case of equirectangular projection. Look around by dragging the direction of view with the mouse with left mouse button pressed.
[[File:Hugin result in aframe.png|center|300px|Equirectangular image used in spheric image in a browser ]]
&lt;center&gt;
[https://niebert.github.io/HuginSample/ Preview of the equirectangular projection in AFrame] - Drag the preview of the equirectangular image with your mouse button pressed.
&lt;/center&gt;

=== Example of the equirectangular image ===  
Preview the [https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg equirectangular JPG-image] and explore the distortion of the image at the top and the bottom of the rectangular JPEG image. 

=== Navigation with multiple equirectangular images ===
The [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html AFrame Navigation example from the river Rhine in Germany] allows to jump from one location at the river rhine to another location at to the equirectangular preview.
 
[[File:river_rhine_spheric_preview.png|center|300px| - Use case of equirectangular projection  - River Rhine example with multiple locations]]

== Subtopics ==
* [[/Maps and Distortion/]]

== Use-Case ==
From a set of standard images of the camera covering a full the from center point of view. In general you cover 360 degree circle with rectangular standard images and you take image with camera for the sky and the floor (preferred without seeing the tripod that might be used for the other images). These set of images are aggregate in one equirectangular image representing a [https://niebert.github.io/HuginSample/ full spheric panorama image], that be be viewed e.g. in [https://aframe.io/sky Aframe] or other panoramic OpenSource viewes, that support equirectangular images. 

=== EQUI2SPH Projection ===
The underlying type of projection is an equirectangular projection EQUI2SPH, that is used e.g in geographical context, where a sphere is projected to rectangular plane on the map. First of all we explore the use-case in [[3D Modelling]] about the spherical use of an spherical panorama image.

== Origin of Terminology and History ==
The ''equirectangular projection'' (also called the ''equidistant cylindrical projection''), and which includes the special case of the ''plate carrée projection'' (also called the ''geographic projection'', ''lat/lon projection'', or ''plane chart''), is a simple [[w:en:map projection|map projection]] attributed to [[w:en:Marinus of Tyre|Marinus of Tyre]], who [[w:en:Ptolemy|Ptolemy]] claims invented the projection about AD 100.&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;5–8, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; The projection maps [[w:en:meridian (geography)|meridians]] to vertical straight lines of constant spacing (for meridional intervals of constant spacing), and [[w:en:circle of latitude|circles of latitude]] to horizontal straight lines of constant spacing (for constant intervals of [[w:en:circle of latitude|parallels]]). The projection is neither [[w:en:equal-area map|equal area]] nor [[w:en:conformal map projection|conformal]]. 

=== Implications of Distortion for Navigation ===
Because of the distortions introduced by this projection, it has little use in [[w:en:navigation|navigation]] or [[w:en:cadastral|cadastral]] mapping and finds its main use in [[w:en:thematic map|thematic mapping]]. 

=== Application in global Raster Datasets ===
In particular, the plate carrée has become a standard for global [[w:en:geographic information system|raster datasets]], such as [[w:en:Celestia|Celestia]], [[w:en:NASA World Wind|NASA World Wind]], the [[w:en:USGS|USGS]] [[w:en:Astrogeology Research Program|Astrogeology Research Program]], and [[w:en:Natural Earth|Natural Earth]], because of the particularly simple relationship between the position of an [[w:en:pixel|image pixel]] on the map and its corresponding geographic location on Earth or other spherical solar system bodies. 

=== Application in panoramic photography ===
In addition it is frequently used in panoramic photography to represent a spherical panoramic image.&lt;ref&gt;{{cite web |title=Equirectangular Projection - PanoTools.org Wiki |url=https://wiki.panotools.org/Equirectangular_Projection |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

== Definition - Equirectangular Projection ==
* '''([[/SPH2EQUI/]])''' The forward projection transforms spherical coordinates into planar coordinates of the equirectangular projection. 
* '''([[/EQUI2SPH/]])''' The reverse projection transforms equirectangular coordinates from the plane back onto the sphere. The formulae presume a [[w:en:figure of the Earth|spherical model]] 

=== Spherical - Longitude and Latitude - SPH ===
* Longitude &lt;math&gt;\lambda \in [ -180^\circ , +180^\circ ] &lt;/math&gt;
* Latitude  &lt;math&gt;\phi \in [ -90^\circ , +90^\circ ]  &lt;/math&gt;

=== Visualization ===
A perspective view of the Earth showing how latitude (&lt;math&gt;\phi&lt;/math&gt;) and longitude (&lt;math&gt;\lambda&lt;/math&gt;) are defined on a spherical model. The graticule spacing is 10 degrees.
[[File:latitude and longitude graticule on a sphere.svg|center|350px|A perspective view of the Earth showing how latitude and longitude]]

==== Longitude - SPH ====
Longitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Greenich Meridian as the [[w:en:Prime Meridian|Prime Meridian]] and is ranging to &lt;math&gt;+180^o&lt;/math&gt; eastward and &lt;math&gt;-180^o&lt;/math&gt; westward. The Greek letter &lt;math&gt;\lambda&lt;/math&gt; (lambda)&lt;ref&gt;{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|title=Coordinate Conversion|website=colorado.edu|access-date=14 March 2018|archive-url=https://web.archive.org/web/20090929121405/http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|archive-date=29 September 2009|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;&quot;&lt;math&gt;\lambda&lt;/math&gt; = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign).&quot;&lt;br /&gt;John P. Snyder, ''[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 Map Projections, A Working Manual] {{Webarchive|url=https://web.archive.org/web/20100701103721/http://pubs.er.usgs.gov/usgspubs/pp/pp1395 |date=2010-07-01 }}'', [[w:en:USGS|USGS]] Professional Paper 1395, page ix&lt;/ref&gt; is used to denote the location of a place on Earth east or west of the Prime Meridian

==== Latitude - SPH ====
Latitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Equator and is ranging to &lt;math&gt;+90^o&lt;/math&gt; towards the North Pole and &lt;math&gt;-90^o&lt;/math&gt; towards the South Pole. The Greek letter &lt;math&gt;\phi&lt;/math&gt; or &lt;math&gt;\varphi&lt;/math&gt; (phi) denotes that angle.

==== Mnemonic - Greek Letter - Phi====
There are two different notations of the greek letter &lt;math&gt; \phi &lt;/math&gt; and &lt;math&gt; \varphi &lt;/math&gt;. In this learning resource the notation &lt;math&gt; \phi &lt;/math&gt; is used to indicate that it denotes the angle at circle that intersects with the North Pole and the South Pole.

=== Definition of Spherical Variables ===
The projections are [[w:en:Function_(mathematics)|mathematical function/mappings]]. For definition of these projections the following variables are defined:
*&lt;math&gt;\lambda&lt;/math&gt; is the [[w:en:longitude|longitude]] of the location to project;
* &lt;math&gt;\phi&lt;/math&gt; is the [[w:en:latitude|latitude]] of the location to project;
* &lt;math&gt;\phi_1&lt;/math&gt; are the standard parallels (north and south of the equator) where the scale of the projection is true;
* &lt;math&gt;\phi_0&lt;/math&gt; is the central parallel of the map (e.g. &lt;math&gt;\phi_0 = 0^\circ &lt;/math&gt; equator);
* &lt;math&gt;\lambda_0&lt;/math&gt; is the central meridian of the map;
* &lt;math&gt;R&lt;/math&gt; is the radius of the globe.
Longitude and latitude variables are defined here in terms of radians.

=== Definition of Equirectangular Planar Variables - EQUI ===
* &lt;math&gt;x_e&lt;/math&gt; is the horizontal coordinate of the projected location on the map;
* &lt;math&gt;y_e&lt;/math&gt; is the vertical coordinate of the projected location on the map;

=== Forward Projection - Spherical to Planar - SPH2EQUI ===
&lt;math&gt;\begin{align}
x &amp;= R \cdot (\lambda - \lambda_0) \cdot \cos (\phi_1)\\
y &amp;= R \cdot (\phi - \phi_0)
\end{align}&lt;/math&gt;

=== Special Case - Forward Projection ===
The {{lang|fr|plate carrée}} ([[w:en:French language|French]], for ''flat square''),&lt;ref&gt;{{Cite web |title=Plate Carrée - a simple example |last=Farkas |first=Gábor |work=O’Reilly Online Learning |date= |access-date=31 December 2022 |url= https://www.oreilly.com/library/view/practical-gis/9781787123328/Text/b21938a9-09f7-46fa-b905-58a0a4ed7d8f.xhtml}}&lt;/ref&gt; is the special case where &lt;math&gt;\varphi_1&lt;/math&gt; is zero. This projection maps ''x'' to be the value of the longitude and ''y'' to be the value of the latitude,&lt;ref&gt;{{cite book |url=https://books.google.co.uk/books?id=-FbVI-2tSuYC&amp;pg=PA119 |p=119 |title=Geographic Information Systems and Science |author1=Paul A. Longley |author2=Michael F. Goodchild |author3=David J. Maguire |author4=David W. Rhind |publisher=John Wiley &amp; Sons |year=2005}}&lt;/ref&gt; and therefore is sometimes called the latitude/longitude or lat/lon(g) projection. 

When the &lt;math&gt;\phi_1&lt;/math&gt; is not zero, such as [[w:en:Marinus of Tyre|Marinus]]'s &lt;math&gt;\phi_1=36&lt;/math&gt;,&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;7, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; or [[w:en:Royal Scottish Geographical Society|Ronald Miller]]'s &lt;math&gt;\phi_1=(37.5, 43.5, 50.5)&lt;/math&gt;,&lt;ref&gt;{{cite web |title=Equidistant Cylindrical (Plate Carrée) |url=https://proj.org/operations/projections/eqc.html |website=PROJ coordinate transformation software library |access-date=25 August 2020}}&lt;/ref&gt; the projection can portray particular latitudes of interest at true scale.

=== Remarks - Ellipsoidal Model ===
While a projection with equally spaced parallels is possible for an '''ellipsoidal model''', it would no longer be equidistant because the distance between parallels on an ellipsoid is not constant. More complex formulae can be used to create an equidistant map whose parallels reflect the true spacing.

===Reverse - Planar to Spherical - EQUI2SPH ===
&lt;math&gt;\begin{align}
\lambda &amp;= \frac{x} {R \cdot \cos (\phi_1)} + \lambda_0\\
\phi &amp;= \frac{y} {R} + \phi_0
\end{align}&lt;/math&gt;

=== Alternative names ===
In spherical panorama viewers, usually:

* &lt;math&gt;\lambda&lt;/math&gt; is called &quot;yaw&quot;;&lt;ref&gt;{{cite web |title=Yaw - PanoTools.org Wiki |url=https://wiki.panotools.org/Yaw |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;
* &lt;math&gt;\phi&lt;/math&gt; is called &quot;pitch&quot;;&lt;ref&gt;{{cite web |title=Pitch - PanoTools.org Wiki |url=https://wiki.panotools.org/Pitch |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

where both are defined in degrees.

== Learning Activities ==
The following learning activities address [[w:en:Projective geometry|Projective Geometry]] from a standard snapshot taken with your camera onto an area on the sphere according to the equirectangular projection. Keep in mind to distinguish between the following projective converters:
*  '''(SPH2EQUI / EQUI2SPH):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x_e,y_e)&lt;/math&gt; coordinates of an equirectangular image
*  '''(SPH2IMG / IMG2SPH):''' &lt;math&gt;(x_e,y_e)&lt;/math&gt; equirectangular coordinates &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.
*  '''(EQUI2IMG / IMG2EQUI):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.

=== Angle of View ===
For the learning activities it is important to understand the 
* '''(HAV)''' Horizontal Angle of View in landscape format and 
* '''(VAV)''' Vertical Angle of View in landscape format. 

=== Learning Task - Angle of View ===
The HAV and VAV differ from camera to camera. Explain why the HAV and VAV are relevant for the calculation of IMG2EQUI projection. Use the following figure to explain the requirements for projection.

=== Visualization of Angle of View ===
The following figure depicts besides the Horizontal (HAV) and Vertical Angle of View (VAV) also the Diagonal Angle of View (DAV).

[[File:Angle of view.svg|350px|center|Horizontal, vertical and diagonal angle of view]]

=== Remark - IMG2EQUI Projection ===
The current projection type of the learning activities is the (IMG2EQUI) projection between the &lt;math&gt;(x,y)&lt;/math&gt; coordinates of a standard image  and the distorted &lt;math&gt;(x_e,y_e)&lt;/math&gt;  coordinates of an equirectangular image and vice verca.

=== HAV and VAV as camera specifc properties ===
Due to the fact that different cameras have different angles of view (HAV and VAV) the visible area of the standard image (taken with your smartphone) might vary. The following animation shows different horizontal angles of view (HAV) e.g. for taking a snapshot vertically upwards towards the sky or blue ceiling.

[[File:Horizontal angle of view.gif|350px|center|dynamic visualization of the Horizontal Angle of View]]

=== IMG2EQUI Projection - Polar Regions ===
In this learning step we take pictures with a standard camera or smartphone and want to calculate the ''(Latitude,Longitude)'' coordinates of the sphere from ''(x,y)'' coordinates of standard image created with your smartphone or camera.
In the first learning step we consider the sphere at the polar regions. These regions are the most distorted areas in the equirectangular projections. 


=== IMG2SPH Projection - Take Snapshots for Learning Task ===
* Take your mobile phone and take two snapshots vertical down and vertical up from your floor and from the ceiling. Select an position in your room where the top and the floor has some visible elements (e.g. a lamp, wooden decorative elements,  ...). Alternatively you can create the two images outside with a cloudy sky and objects lying on the ground. 
* select a center of a circle and a radius that fits into both images (maximize the radius of the circle,
* in this module we will project these two images to the polar regions of the sphere within the rectangular coordinate system of the image.

==== Initial State - Image Sky ====
The following image shows the initial state taking a picture from the sky. The camera is located in the center of the red sphere and takes an image of the blue sky or the ceiling in a room. The ceiling is visualized as a blue plane. The learning task is to calculate the angles for the corresponding point on the sphere. 
 
[[File:Circle plane equirectangular.png|350px|Equirectangular Projection - Sky Image - Polar Projection]]

==== Side view with polar image ====
The side view can be used to derive the calculation of the green angle with &lt;math&gt;\alpha:=\arctan(x)&lt;/math&gt;. The relevant angle for the equirectangular projection of the polar region is marked red. 

[[File:Equirectangular polar side view.gif|350px|Equirectangular projection side view - relevant angle is the red angle]]

==== Used Part of Snapshot with a Camera ====
The first image is snapshot with a camera taking a picture vertically upwards to the ceiling (into the sky). The red circle is the projected area from the image (sky/ceiling) onto a part of the equirectangular image visualized in the next section of the learning resource.

[[File:Equirectangular polar image.svg|350px|Equirectangular source image with circle to be projected]]

==== Equirectangular Projection of Circle - Rectangle ====
Explain, why the circle in the source image of the polar region (North Pole) is a rectangle after projection in the equirectangular image.

[[File:Equirectangular polar image projection.svg|350px|Equirectangular source image with circle to be projected]]

==== Learning Task - Function for Coordinate Transformation IMG2EQUI ====
We denote by &lt;math&gt;P_{IMG}(x,y)&lt;/math&gt; the color information of a pixel in the source image (IMG) and with &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt;  the color information of the pixel in the image in the destination format (EQUI). Define the function &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt; by calculating the corresponding &lt;math&gt;(x,y)&lt;/math&gt; coordinates in the IMG source image. This calculation is used to set pixels in the destination format EQUI by setting  
:&lt;math display=&quot;block&quot;&gt;P_{EQUI}(x_e,y_e):=P_{IMG}(x,y).&lt;/math&gt;
'''Remark:''' You need basic knowledge in [[trigonometry]] to perform this learning task.

==== Coordinate System of Graphics ====
[[File:Graphics coordinate system.svg|thumb|Coordinate system in Graphics]]
The coordinate system of an image has a different orientation in y-axis. This is shown in the diagram.
* &lt;math&gt;x_{max}&lt;/math&gt; is the maximal value of the on the x-axis of the image
* &lt;math&gt;y_{max}&lt;/math&gt; is the maximal value of the on the y-axis  the of the image
* The origin of the coordinate system is on the top left. 
Keep this in mind, when you use the coordinate system for your experiments with projections (see also equirectangular projection.

==== Rectangular to Sphere - Projection ====

[[File:Equirectangular ceiling floor.jpg|thumb|Equirectangular projection with marked ceiling and floor]]
Now we use a the standard rectangular image on the right as in input for the equirectangular projection on the sphere and we view the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image in panoramic preview on the sphere]. What can be observed, if you analyze the projected area of red [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg ceiling and a marked green rectangle on the floor].
* Drag the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image upwards to the ceiling] 
* and [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg downwards to floor].

==== Screenshots - Projection on Sphere of the standard Image ====
{| class=&quot;wikitable&quot;
|+ Screenshot of projection
|-
! Ceiling/Sky !! Floor
|-
| [[File:Equirectangular screenshot ceiling.png|320px|Screenshot ceiling/sky of standard image projected on a sphere with equirectangular projection]] || [[File:Equirectangular screenshot floor.png|320px|Screenshot floor of standard image projected on a sphere with equirectangular projection]]
|}

==== Learning Task - Projection of Sky in Graphics Coordinate System ====

[[File:Equirectangular coordinate system sky.svg|thumb|Equirectangular coordinate system sky / ceiling]]

Graphics have an own coordinate system to display points in pixel graphics or geometric objects like lines, polygons and circles in the coordinate system. The diagram shows the coordinate system of the image. Keep in mind that the coordinate of the y-axis has a different orientation than the y-axis in the standard 2D [[w:de:Cartesian coordinate system|Cartesian coordinate system]]. This information is relevant if you experiment with equirectangular projections in [[Projective Geometry Playground]].

Calculate the &lt;math&gt;y_{sky}&lt;/math&gt; for a specific angle of view of your camera.

==== Learning Task - Calculation ====
[[File:Horizontal angle of view.jpg|thumb|Fullscreen image of Door for HAV/VAV calc
[[File:Horizontal angle of view meaure distance.jpg|thumb|Measure distance from camera position to door ]]
ulation]]

Calculate the equirectangular projection for any &lt;math&gt;(x,y)&lt;/math&gt; point in the red circle of the source image IMG into the destination format EQUI with:
* '''LibreOffice - HAV/VAV:''' Take image of a door and calculate the horizontal and vertical angle of view (HAV and VAV)
* '''LibreOffice - Coordinates:''' Create a Spreadsheet document for the calculation of coordinates from a given coordinate in the equirectangular image the corresponding coordinates in the source image of your camera.
* [[Projective Geometry Playground|Javascript and HTML canvas]] - (see [[Projective Geometry Playground]]),
* [[w:en:GNU Octave|Octave]] with [https://gnu-octave.github.io/packages/image/ image-package]&lt;ref&gt; Carnë Draug,
Hartmut Gimpel, Avinoam Kalma (2022) Image Package Octave URL: https://gnu-octave.github.io/packages/image/ (March, 28th, 2024)&lt;/ref&gt;, or 
* Python with [https://github.com/python-pillow/Pillow Image Processing Library pillow]  by Jeffrey A. Clark   
Select an implementation of your choice. LibreOffice has the minimal requirements on programming skill but only coordinate transformation can be performed without a visual output.

=== Learning Task - Sky - North Pole ===
[[File:Sky image for equirectangular projection.jpg|thumb|Sky - equirectangular projection - north pole]]
A sky image can be used to project a circular area in the source image to the rectangular part at the top of the generated equirectangular image. 

Import the equirectangular projection in Aframe to preview the result.

==== Learning Task - Floor - South Pole ====
[[File:Floor sand for equirectangular.jpg|thumb|Floor Image - sand as demo input for equirectangular projection]]Transfer the lesson learned from north pole to the south pole and project the beach image as floor to the south pole. 
* What are differences and similarities between both projections?


== Final Result ==
[[File:Aldara parks.jpg|thumb|[https://niebert.github.io/HuginSample/Aldara_parks.html Equirectangular Image from Wikiversity used for Aframe 360 Degree Image] (see [[3D Modelling/Create 3D Models/Hugin|Hugin]])]]
{{PanoViewer|Frary Dining Hall 360-degree view.jpg|Dining Hall 360-degree view - with PanoViewer Template}}
* [https://niebert.github.io/HuginSample/ Final Results]  would can be previewed in Aframe (see [https://www.github.com/HuginSample HuginSample Files on Github]) or with the Wiki,
* [https://niebert.github.com/HuginSample/Aldara_parks.html Aldara Parks 360 Degree Image in Aframe] with an already uploaded equirectangular image in WikiMedia Commons by U.Bardins
* [https://panoviewer.toolforge.org/#Aldara_parks.jpg PanoViewer 360 degree view of Aldara Parks image]]

==See also==
* [[Projective Geometry Playground]]
* [[GeoGebra/Perspective Drawing on Mirror|Perspective Drawing on Mirror]]
* [[3D Modelling]]
* [[w:en:Cartography|Cartography]]
* [[w:en:Cassini projection|Cassini projection]]
* [[w:en:Gall–Peters projection|Gall–Peters projection]] with resolution regarding the use of rectangular world maps
* [[w:en:List of map projections|List of map projections]]
* [[w:en:Mercator projection|Mercator projection]]
* [[w:en:360 video projection|360 video projection]]
* [[3D_Modelling/Examples/Panorama_360|3D Modelling - 360 degree panorama]]
* [[Geometry]]
* [[w:en:Projective_geometry|Projective Geometry]]
* [[Portal:Mathematics]]
* [[w:en:GNU Octave|Octave]]
* [[Geogebra]]
* [[b:en:Javascript|Wikibook: Javascript]]
* [[Portal:Mathematics]]

==References==
{{Reflist}}

==External links==
* [https://visibleearth.nasa.gov/view.php?id=57730 Global MODIS based satellite map] The blue marble: land surface, ocean color, and sea ice.
* [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net.
* [http://wiki.panotools.org/Equirectangular Panoramic Equirectangular Projection], PanoTools wiki.
* [https://proj4.org/operations/projections/eqc.html Equidistant Cylindrical (Plate Carrée) in proj4]

== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/3D%20Modelling 3D Modelling]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Equirectangular%20projection https://en.wikiversity.org/wiki/Equirectangular%20projection]
--&gt;
* [https://en.wikiversity.org/wiki/Equirectangular%20projection This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
* Source: Wikiversity  https://en.wikiversity.org/wiki/Equirectangular%20projection
* see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal].
&lt;!-- * Next contents of the course are [[]] --&gt;;

=== Wikipedia2Wikiversiy===
This page was based on the following [https://en.wikipedia.org/wiki/Equirectangular%20projection wikipedia-source page]:
* [https://en.wikipedia.org/wiki/Equirectangular%20projection Equirectangular projection] https://en.wikipedia.org/wiki/Equirectangular%20projection
* Datum: 1/9/2023
* [https://niebert.github.io/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.io/Wikipedia2Wikiversity


[[Category:Map projections]]
[[Category:Equidistant projections]]
[[Category:Cylindrical projections]]
[[Category:Wiki2Reveal]]</text>
      <sha1>pkx7tgvggkzv61ftc7diowyvqc91t05</sha1>
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      <text bytes="26765" xml:space="preserve">== Wiki2Reveal ==
This learning resource can be used as [[Wiki2Reveal]] slides in mathematics courses as introduction to [[w:en:Projective geometry|projective geometry]].

* Start '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=Geometry&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=Geometry Wiki2Reveal]''' [[File:Wiki2Reveal Logo.png|35px]]
  

== Introduction == 
In this learning resource the generation of an equirectangular projection is the objective. The learning resource is use-case driven, [https://niebert.github.io/HuginSample/ AFrame Example Durlach] is the first use-case of equirectangular projection. Look around by dragging the direction of view with the mouse with left mouse button pressed.
[[File:Hugin result in aframe.png|center|300px|Equirectangular image used in spheric image in a browser ]]
&lt;center&gt;
[https://niebert.github.io/HuginSample/ Preview of the equirectangular projection in AFrame] - Drag the preview of the equirectangular image with your mouse button pressed.
&lt;/center&gt;

=== Example of the equirectangular image ===  
Preview the [https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg equirectangular JPG-image] and explore the distortion of the image at the top and the bottom of the rectangular JPEG image. 

=== Navigation with multiple equirectangular images ===
The [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html AFrame Navigation example from the river Rhine in Germany] allows to jump from one location at the river rhine to another location at to the equirectangular preview.
 
[[File:river_rhine_spheric_preview.png|center|300px| - Use case of equirectangular projection  - River Rhine example with multiple locations]]

== Subtopics ==
* [[/Maps and Distortion/]]

== Use-Case ==
From a set of standard images of the camera covering a full the from center point of view. In general you cover 360 degree circle with rectangular standard images and you take image with camera for the sky and the floor (preferred without seeing the tripod that might be used for the other images). These set of images are aggregate in one equirectangular image representing a [https://niebert.github.io/HuginSample/ full spheric panorama image], that be be viewed e.g. in [https://aframe.io/sky Aframe] or other panoramic OpenSource viewes, that support equirectangular images. 

=== EQUI2SPH Projection ===
The underlying type of projection is an equirectangular projection EQUI2SPH, that is used e.g in geographical context, where a sphere is projected to rectangular plane on the map. First of all we explore the use-case in [[3D Modelling]] about the spherical use of an spherical panorama image.

== Origin of Terminology and History ==
The ''equirectangular projection'' (also called the ''equidistant cylindrical projection''), and which includes the special case of the ''plate carrée projection'' (also called the ''geographic projection'', ''lat/lon projection'', or ''plane chart''), is a simple [[w:en:map projection|map projection]] attributed to [[w:en:Marinus of Tyre|Marinus of Tyre]], who [[w:en:Ptolemy|Ptolemy]] claims invented the projection about AD 100.&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;5–8, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; The projection maps [[w:en:meridian (geography)|meridians]] to vertical straight lines of constant spacing (for meridional intervals of constant spacing), and [[w:en:circle of latitude|circles of latitude]] to horizontal straight lines of constant spacing (for constant intervals of [[w:en:circle of latitude|parallels]]). The projection is neither [[w:en:equal-area map|equal area]] nor [[w:en:conformal map projection|conformal]]. 

=== Implications of Distortion for Navigation ===
Because of the distortions introduced by this projection, it has little use in [[w:en:navigation|navigation]] or [[w:en:cadastral|cadastral]] mapping and finds its main use in [[w:en:thematic map|thematic mapping]]. 

=== Application in global Raster Datasets ===
In particular, the plate carrée has become a standard for global [[w:en:geographic information system|raster datasets]], such as [[w:en:Celestia|Celestia]], [[w:en:NASA World Wind|NASA World Wind]], the [[w:en:USGS|USGS]] [[w:en:Astrogeology Research Program|Astrogeology Research Program]], and [[w:en:Natural Earth|Natural Earth]], because of the particularly simple relationship between the position of an [[w:en:pixel|image pixel]] on the map and its corresponding geographic location on Earth or other spherical solar system bodies. 

=== Application in panoramic photography ===
In addition it is frequently used in panoramic photography to represent a spherical panoramic image.&lt;ref&gt;{{cite web |title=Equirectangular Projection - PanoTools.org Wiki |url=https://wiki.panotools.org/Equirectangular_Projection |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

== Definition - Equirectangular Projection ==
* '''([[/SPH2EQUI/]])''' The forward projection transforms spherical coordinates into planar coordinates of the equirectangular projection. 
* '''([[/EQUI2SPH/]])''' The reverse projection transforms equirectangular coordinates from the plane back onto the sphere. The formulae presume a [[w:en:figure of the Earth|spherical model]] 

=== Spherical - Longitude and Latitude - SPH ===
* Longitude &lt;math&gt;\lambda \in [ -180^\circ , +180^\circ ] &lt;/math&gt;
* Latitude  &lt;math&gt;\phi \in [ -90^\circ , +90^\circ ]  &lt;/math&gt;

=== Visualization ===
A perspective view of the Earth showing how latitude (&lt;math&gt;\phi&lt;/math&gt;) and longitude (&lt;math&gt;\lambda&lt;/math&gt;) are defined on a spherical model. The graticule spacing is 10 degrees.
[[File:latitude and longitude graticule on a sphere.svg|center|350px|A perspective view of the Earth showing how latitude and longitude]]

==== Longitude - SPH ====
Longitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Greenich Meridian as the [[w:en:Prime Meridian|Prime Meridian]] and is ranging to &lt;math&gt;+180^o&lt;/math&gt; eastward and &lt;math&gt;-180^o&lt;/math&gt; westward. The Greek letter &lt;math&gt;\lambda&lt;/math&gt; (lambda)&lt;ref&gt;{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|title=Coordinate Conversion|website=colorado.edu|access-date=14 March 2018|archive-url=https://web.archive.org/web/20090929121405/http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|archive-date=29 September 2009|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;&quot;&lt;math&gt;\lambda&lt;/math&gt; = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign).&quot;&lt;br /&gt;John P. Snyder, ''[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 Map Projections, A Working Manual] {{Webarchive|url=https://web.archive.org/web/20100701103721/http://pubs.er.usgs.gov/usgspubs/pp/pp1395 |date=2010-07-01 }}'', [[w:en:USGS|USGS]] Professional Paper 1395, page ix&lt;/ref&gt; is used to denote the location of a place on Earth east or west of the Prime Meridian

==== Latitude - SPH ====
Latitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Equator and is ranging to &lt;math&gt;+90^o&lt;/math&gt; towards the North Pole and &lt;math&gt;-90^o&lt;/math&gt; towards the South Pole. The Greek letter &lt;math&gt;\phi&lt;/math&gt; or &lt;math&gt;\varphi&lt;/math&gt; (phi) denotes that angle.

==== Mnemonic - Greek Letter - Phi====
There are two different notations of the greek letter &lt;math&gt; \phi &lt;/math&gt; and &lt;math&gt; \varphi &lt;/math&gt;. In this learning resource the notation &lt;math&gt; \phi &lt;/math&gt; is used to indicate that it denotes the angle at circle that intersects with the North Pole and the South Pole.

=== Definition of Spherical Variables ===
The projections are [[w:en:Function_(mathematics)|mathematical function/mappings]]. For definition of these projections the following variables are defined:
*&lt;math&gt;\lambda&lt;/math&gt; is the [[w:en:longitude|longitude]] of the location to project;
* &lt;math&gt;\phi&lt;/math&gt; is the [[w:en:latitude|latitude]] of the location to project;
* &lt;math&gt;\phi_1&lt;/math&gt; are the standard parallels (north and south of the equator) where the scale of the projection is true;
* &lt;math&gt;\phi_0&lt;/math&gt; is the central parallel of the map (e.g. &lt;math&gt;\phi_0 = 0^\circ &lt;/math&gt; equator);
* &lt;math&gt;\lambda_0&lt;/math&gt; is the central meridian of the map;
* &lt;math&gt;R&lt;/math&gt; is the radius of the globe.
Longitude and latitude variables are defined here in terms of radians.

=== Definition of Equirectangular Planar Variables - EQUI ===
* &lt;math&gt;x_e&lt;/math&gt; is the horizontal coordinate of the projected location on the map;
* &lt;math&gt;y_e&lt;/math&gt; is the vertical coordinate of the projected location on the map;

=== Forward Projection - Spherical to Planar - SPH2EQUI ===
&lt;math&gt;\begin{align}
x &amp;= R \cdot (\lambda - \lambda_0) \cdot \cos (\phi_1)\\
y &amp;= R \cdot (\phi - \phi_0)
\end{align}&lt;/math&gt;

=== Special Case - Forward Projection ===
The {{lang|fr|plate carrée}} ([[w:en:French language|French]], for ''flat square''),&lt;ref&gt;{{Cite web |title=Plate Carrée - a simple example |last=Farkas |first=Gábor |work=O’Reilly Online Learning |date= |access-date=31 December 2022 |url= https://www.oreilly.com/library/view/practical-gis/9781787123328/Text/b21938a9-09f7-46fa-b905-58a0a4ed7d8f.xhtml}}&lt;/ref&gt; is the special case where &lt;math&gt;\varphi_1&lt;/math&gt; is zero. This projection maps ''x'' to be the value of the longitude and ''y'' to be the value of the latitude,&lt;ref&gt;{{cite book |url=https://books.google.co.uk/books?id=-FbVI-2tSuYC&amp;pg=PA119 |p=119 |title=Geographic Information Systems and Science |author1=Paul A. Longley |author2=Michael F. Goodchild |author3=David J. Maguire |author4=David W. Rhind |publisher=John Wiley &amp; Sons |year=2005}}&lt;/ref&gt; and therefore is sometimes called the latitude/longitude or lat/lon(g) projection. 

When the &lt;math&gt;\phi_1&lt;/math&gt; is not zero, such as [[w:en:Marinus of Tyre|Marinus]]'s &lt;math&gt;\phi_1=36&lt;/math&gt;,&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;7, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; or [[w:en:Royal Scottish Geographical Society|Ronald Miller]]'s &lt;math&gt;\phi_1=(37.5, 43.5, 50.5)&lt;/math&gt;,&lt;ref&gt;{{cite web |title=Equidistant Cylindrical (Plate Carrée) |url=https://proj.org/operations/projections/eqc.html |website=PROJ coordinate transformation software library |access-date=25 August 2020}}&lt;/ref&gt; the projection can portray particular latitudes of interest at true scale.

=== Remarks - Ellipsoidal Model ===
While a projection with equally spaced parallels is possible for an '''ellipsoidal model''', it would no longer be equidistant because the distance between parallels on an ellipsoid is not constant. More complex formulae can be used to create an equidistant map whose parallels reflect the true spacing.

===Reverse - Planar to Spherical - EQUI2SPH ===
&lt;math&gt;\begin{align}
\lambda &amp;= \frac{x} {R \cdot \cos (\phi_1)} + \lambda_0\\
\phi &amp;= \frac{y} {R} + \phi_0
\end{align}&lt;/math&gt;

=== Alternative names ===
In spherical panorama viewers, usually:

* &lt;math&gt;\lambda&lt;/math&gt; is called &quot;yaw&quot;;&lt;ref&gt;{{cite web |title=Yaw - PanoTools.org Wiki |url=https://wiki.panotools.org/Yaw |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;
* &lt;math&gt;\phi&lt;/math&gt; is called &quot;pitch&quot;;&lt;ref&gt;{{cite web |title=Pitch - PanoTools.org Wiki |url=https://wiki.panotools.org/Pitch |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

where both are defined in degrees.

== Learning Activities ==
The following learning activities address [[w:en:Projective geometry|Projective Geometry]] from a standard snapshot taken with your camera onto an area on the sphere according to the equirectangular projection. Keep in mind to distinguish between the following projective converters:
*  '''(SPH2EQUI / EQUI2SPH):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x_e,y_e)&lt;/math&gt; coordinates of an equirectangular image
*  '''(SPH2IMG / IMG2SPH):''' &lt;math&gt;(x_e,y_e)&lt;/math&gt; equirectangular coordinates &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.
*  '''(EQUI2IMG / IMG2EQUI):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.

=== Angle of View ===
For the learning activities it is important to understand the 
* '''(HAV)''' Horizontal Angle of View in landscape format and 
* '''(VAV)''' Vertical Angle of View in landscape format. 

=== Learning Task - Angle of View ===
The HAV and VAV differ from camera to camera. Explain why the HAV and VAV are relevant for the calculation of IMG2EQUI projection. Use the following figure to explain the requirements for projection.

=== Visualization of Angle of View ===
The following figure depicts besides the Horizontal (HAV) and Vertical Angle of View (VAV) also the Diagonal Angle of View (DAV).

[[File:Angle of view.svg|350px|center|Horizontal, vertical and diagonal angle of view]]

=== Remark - IMG2EQUI Projection ===
The current projection type of the learning activities is the (IMG2EQUI) projection between the &lt;math&gt;(x,y)&lt;/math&gt; coordinates of a standard image  and the distorted &lt;math&gt;(x_e,y_e)&lt;/math&gt;  coordinates of an equirectangular image and vice verca.

=== HAV and VAV as camera specifc properties ===
Due to the fact that different cameras have different angles of view (HAV and VAV) the visible area of the standard image (taken with your smartphone) might vary. The following animation shows different horizontal angles of view (HAV) e.g. for taking a snapshot vertically upwards towards the sky or blue ceiling.

[[File:Horizontal angle of view.gif|350px|center|dynamic visualization of the Horizontal Angle of View]]

== Polar Region - Distortion ==
The projection creates especially at the &quot;North Pole&quot; and the &quot;South Pole&quot; the [[Mapping and Distortion|heaviest distortion in comparison to distances measured]] on the surface of the sphere. For panoramic views the distortion is just a matter of storage of the spheric pixel information in a rectangular format. On the image of a market place you will see that the panoramic viewes transfer the rectangular images into a natural view where you can look around and explore the location from different angles and with multiple equirectangular images from many locations (see [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html river Rhine example Cologne]).

=== IMG2EQUI Projection - Polar Regions ===
In this learning step we take pictures with a standard camera or smartphone and want to calculate the ''(Latitude,Longitude)'' coordinates of the sphere from ''(x,y)'' coordinates of standard image created with your smartphone or camera.
In the first learning step we consider the sphere at the polar regions. These regions are the most distorted areas in the equirectangular projections.

=== IMG2SPH Projection - Take Snapshots for Learning Task ===
* Take your mobile phone and take two snapshots vertical down and vertical up from your floor and from the ceiling. Select an position in your room where the top and the floor has some visible elements (e.g. a lamp, wooden decorative elements,  ...). Alternatively you can create the two images outside with a cloudy sky and objects lying on the ground. 
* select a center of a circle and a radius that fits into both images (maximize the radius of the circle,
* in this module we will project these two images to the polar regions of the sphere within the rectangular coordinate system of the image.

==== Initial State - Image Sky ====
The following image shows the initial state taking a picture from the sky. The camera is located in the center of the red sphere and takes an image of the blue sky or the ceiling in a room. The ceiling is visualized as a blue plane. The learning task is to calculate the angles for the corresponding point on the sphere. 
 
[[File:Circle plane equirectangular.png|350px|Equirectangular Projection - Sky Image - Polar Projection]]

==== Side view with polar image ====
The side view can be used to derive the calculation of the green angle with &lt;math&gt;\alpha:=\arctan(x)&lt;/math&gt;. The relevant angle for the equirectangular projection of the polar region is marked red. 

[[File:Equirectangular polar side view.gif|350px|Equirectangular projection side view - relevant angle is the red angle]]

==== Used Part of Snapshot with a Camera ====
The first image is snapshot with a camera taking a picture vertically upwards to the ceiling (into the sky). The red circle is the projected area from the image (sky/ceiling) onto a part of the equirectangular image visualized in the next section of the learning resource.

[[File:Equirectangular polar image.svg|350px|Equirectangular source image with circle to be projected]]

==== Equirectangular Projection of Circle - Rectangle ====
Explain, why the circle in the source image of the polar region (North Pole) is a rectangle after projection in the equirectangular image.

[[File:Equirectangular polar image projection.svg|350px|Equirectangular source image with circle to be projected]]

==== Learning Task - Function for Coordinate Transformation IMG2EQUI ====
We denote by &lt;math&gt;P_{IMG}(x,y)&lt;/math&gt; the color information of a pixel in the source image (IMG) and with &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt;  the color information of the pixel in the image in the destination format (EQUI). Define the function &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt; by calculating the corresponding &lt;math&gt;(x,y)&lt;/math&gt; coordinates in the IMG source image. This calculation is used to set pixels in the destination format EQUI by setting  
:&lt;math display=&quot;block&quot;&gt;P_{EQUI}(x_e,y_e):=P_{IMG}(x,y).&lt;/math&gt;
'''Remark:''' You need basic knowledge in [[trigonometry]] to perform this learning task.

==== Coordinate System of Graphics ====
[[File:Graphics coordinate system.svg|thumb|Coordinate system in Graphics]]
The coordinate system of an image has a different orientation in y-axis. This is shown in the diagram.
* &lt;math&gt;x_{max}&lt;/math&gt; is the maximal value of the on the x-axis of the image
* &lt;math&gt;y_{max}&lt;/math&gt; is the maximal value of the on the y-axis  the of the image
* The origin of the coordinate system is on the top left. 
Keep this in mind, when you use the coordinate system for your experiments with projections (see also equirectangular projection.

==== Rectangular to Sphere - Projection ====

[[File:Equirectangular ceiling floor.jpg|thumb|Equirectangular projection with marked ceiling and floor]]
Now we use a the standard rectangular image on the right as in input for the equirectangular projection on the sphere and we view the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image in panoramic preview on the sphere]. What can be observed, if you analyze the projected area of red [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg ceiling and a marked green rectangle on the floor].
* Drag the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image upwards to the ceiling] 
* and [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg downwards to floor].

==== Screenshots - Projection on Sphere of the standard Image ====
{| class=&quot;wikitable&quot;
|+ Screenshot of projection
|-
! Ceiling/Sky !! Floor
|-
| [[File:Equirectangular screenshot ceiling.png|320px|Screenshot ceiling/sky of standard image projected on a sphere with equirectangular projection]] || [[File:Equirectangular screenshot floor.png|320px|Screenshot floor of standard image projected on a sphere with equirectangular projection]]
|}

==== Learning Task - Projection of Sky in Graphics Coordinate System ====

[[File:Equirectangular coordinate system sky.svg|thumb|Equirectangular coordinate system sky / ceiling]]

Graphics have an own coordinate system to display points in pixel graphics or geometric objects like lines, polygons and circles in the coordinate system. The diagram shows the coordinate system of the image. Keep in mind that the coordinate of the y-axis has a different orientation than the y-axis in the standard 2D [[w:de:Cartesian coordinate system|Cartesian coordinate system]]. This information is relevant if you experiment with equirectangular projections in [[Projective Geometry Playground]].

Calculate the &lt;math&gt;y_{sky}&lt;/math&gt; for a specific angle of view of your camera.

==== Learning Task - Calculation ====
[[File:Horizontal angle of view.jpg|thumb|Fullscreen image of Door for HAV/VAV calc
[[File:Horizontal angle of view meaure distance.jpg|thumb|Measure distance from camera position to door ]]
ulation]]

Calculate the equirectangular projection for any &lt;math&gt;(x,y)&lt;/math&gt; point in the red circle of the source image IMG into the destination format EQUI with:
* '''LibreOffice - HAV/VAV:''' Take image of a door and calculate the horizontal and vertical angle of view (HAV and VAV)
* '''LibreOffice - Coordinates:''' Create a Spreadsheet document for the calculation of coordinates from a given coordinate in the equirectangular image the corresponding coordinates in the source image of your camera.
* [[Projective Geometry Playground|Javascript and HTML canvas]] - (see [[Projective Geometry Playground]]),
* [[w:en:GNU Octave|Octave]] with [https://gnu-octave.github.io/packages/image/ image-package]&lt;ref&gt; Carnë Draug,
Hartmut Gimpel, Avinoam Kalma (2022) Image Package Octave URL: https://gnu-octave.github.io/packages/image/ (March, 28th, 2024)&lt;/ref&gt;, or 
* Python with [https://github.com/python-pillow/Pillow Image Processing Library pillow]  by Jeffrey A. Clark   
Select an implementation of your choice. LibreOffice has the minimal requirements on programming skill but only coordinate transformation can be performed without a visual output.

=== Learning Task - Sky - North Pole ===
[[File:Sky image for equirectangular projection.jpg|thumb|Sky - equirectangular projection - north pole]]
A sky image can be used to project a circular area in the source image to the rectangular part at the top of the generated equirectangular image. 

Import the equirectangular projection in Aframe to preview the result.

==== Learning Task - Floor - South Pole ====
[[File:Floor sand for equirectangular.jpg|thumb|Floor Image - sand as demo input for equirectangular projection]]Transfer the lesson learned from north pole to the south pole and project the beach image as floor to the south pole. 
* What are differences and similarities between both projections?


== Final Result ==
[[File:Aldara parks.jpg|thumb|[https://niebert.github.io/HuginSample/Aldara_parks.html Equirectangular Image from Wikiversity used for Aframe 360 Degree Image] (see [[3D Modelling/Create 3D Models/Hugin|Hugin]])]]
{{PanoViewer|Frary Dining Hall 360-degree view.jpg|Dining Hall 360-degree view - with PanoViewer Template}}
* [https://niebert.github.io/HuginSample/ Final Results]  would can be previewed in Aframe (see [https://www.github.com/HuginSample HuginSample Files on Github]) or with the Wiki,
* [https://niebert.github.com/HuginSample/Aldara_parks.html Aldara Parks 360 Degree Image in Aframe] with an already uploaded equirectangular image in WikiMedia Commons by U.Bardins
* [https://panoviewer.toolforge.org/#Aldara_parks.jpg PanoViewer 360 degree view of Aldara Parks image]]

==See also==
* [[Projective Geometry Playground]]
* [[GeoGebra/Perspective Drawing on Mirror|Perspective Drawing on Mirror]]
* [[3D Modelling]]
* [[w:en:Cartography|Cartography]]
* [[w:en:Cassini projection|Cassini projection]]
* [[w:en:Gall–Peters projection|Gall–Peters projection]] with resolution regarding the use of rectangular world maps
* [[w:en:List of map projections|List of map projections]]
* [[w:en:Mercator projection|Mercator projection]]
* [[w:en:360 video projection|360 video projection]]
* [[3D_Modelling/Examples/Panorama_360|3D Modelling - 360 degree panorama]]
* [[Geometry]]
* [[w:en:Projective_geometry|Projective Geometry]]
* [[Portal:Mathematics]]
* [[w:en:GNU Octave|Octave]]
* [[Geogebra]]
* [[b:en:Javascript|Wikibook: Javascript]]
* [[Portal:Mathematics]]

==References==
{{Reflist}}

==External links==
* [https://visibleearth.nasa.gov/view.php?id=57730 Global MODIS based satellite map] The blue marble: land surface, ocean color, and sea ice.
* [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net.
* [http://wiki.panotools.org/Equirectangular Panoramic Equirectangular Projection], PanoTools wiki.
* [https://proj4.org/operations/projections/eqc.html Equidistant Cylindrical (Plate Carrée) in proj4]

== Page Information ==
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=== Wikipedia2Wikiversiy===
This page was based on the following [https://en.wikipedia.org/wiki/Equirectangular%20projection wikipedia-source page]:
* [https://en.wikipedia.org/wiki/Equirectangular%20projection Equirectangular projection] https://en.wikipedia.org/wiki/Equirectangular%20projection
* Datum: 1/9/2023
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[[Category:Map projections]]
[[Category:Equidistant projections]]
[[Category:Cylindrical projections]]
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      <comment>/* Polar Region - Distortion */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="26761" xml:space="preserve">== Wiki2Reveal ==
This learning resource can be used as [[Wiki2Reveal]] slides in mathematics courses as introduction to [[w:en:Projective geometry|projective geometry]].

* Start '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=Geometry&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=Geometry Wiki2Reveal]''' [[File:Wiki2Reveal Logo.png|35px]]
  

== Introduction == 
In this learning resource the generation of an equirectangular projection is the objective. The learning resource is use-case driven, [https://niebert.github.io/HuginSample/ AFrame Example Durlach] is the first use-case of equirectangular projection. Look around by dragging the direction of view with the mouse with left mouse button pressed.
[[File:Hugin result in aframe.png|center|300px|Equirectangular image used in spheric image in a browser ]]
&lt;center&gt;
[https://niebert.github.io/HuginSample/ Preview of the equirectangular projection in AFrame] - Drag the preview of the equirectangular image with your mouse button pressed.
&lt;/center&gt;

=== Example of the equirectangular image ===  
Preview the [https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg equirectangular JPG-image] and explore the distortion of the image at the top and the bottom of the rectangular JPEG image. 

=== Navigation with multiple equirectangular images ===
The [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html AFrame Navigation example from the river Rhine in Germany] allows to jump from one location at the river rhine to another location at to the equirectangular preview.
 
[[File:river_rhine_spheric_preview.png|center|300px| - Use case of equirectangular projection  - River Rhine example with multiple locations]]

== Subtopics ==
* [[/Maps and Distortion/]]

== Use-Case ==
From a set of standard images of the camera covering a full the from center point of view. In general you cover 360 degree circle with rectangular standard images and you take image with camera for the sky and the floor (preferred without seeing the tripod that might be used for the other images). These set of images are aggregate in one equirectangular image representing a [https://niebert.github.io/HuginSample/ full spheric panorama image], that be be viewed e.g. in [https://aframe.io/sky Aframe] or other panoramic OpenSource viewes, that support equirectangular images. 

=== EQUI2SPH Projection ===
The underlying type of projection is an equirectangular projection EQUI2SPH, that is used e.g in geographical context, where a sphere is projected to rectangular plane on the map. First of all we explore the use-case in [[3D Modelling]] about the spherical use of an spherical panorama image.

== Origin of Terminology and History ==
The ''equirectangular projection'' (also called the ''equidistant cylindrical projection''), and which includes the special case of the ''plate carrée projection'' (also called the ''geographic projection'', ''lat/lon projection'', or ''plane chart''), is a simple [[w:en:map projection|map projection]] attributed to [[w:en:Marinus of Tyre|Marinus of Tyre]], who [[w:en:Ptolemy|Ptolemy]] claims invented the projection about AD 100.&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;5–8, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; The projection maps [[w:en:meridian (geography)|meridians]] to vertical straight lines of constant spacing (for meridional intervals of constant spacing), and [[w:en:circle of latitude|circles of latitude]] to horizontal straight lines of constant spacing (for constant intervals of [[w:en:circle of latitude|parallels]]). The projection is neither [[w:en:equal-area map|equal area]] nor [[w:en:conformal map projection|conformal]]. 

=== Implications of Distortion for Navigation ===
Because of the distortions introduced by this projection, it has little use in [[w:en:navigation|navigation]] or [[w:en:cadastral|cadastral]] mapping and finds its main use in [[w:en:thematic map|thematic mapping]]. 

=== Application in global Raster Datasets ===
In particular, the plate carrée has become a standard for global [[w:en:geographic information system|raster datasets]], such as [[w:en:Celestia|Celestia]], [[w:en:NASA World Wind|NASA World Wind]], the [[w:en:USGS|USGS]] [[w:en:Astrogeology Research Program|Astrogeology Research Program]], and [[w:en:Natural Earth|Natural Earth]], because of the particularly simple relationship between the position of an [[w:en:pixel|image pixel]] on the map and its corresponding geographic location on Earth or other spherical solar system bodies. 

=== Application in panoramic photography ===
In addition it is frequently used in panoramic photography to represent a spherical panoramic image.&lt;ref&gt;{{cite web |title=Equirectangular Projection - PanoTools.org Wiki |url=https://wiki.panotools.org/Equirectangular_Projection |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

== Definition - Equirectangular Projection ==
* '''([[/SPH2EQUI/]])''' The forward projection transforms spherical coordinates into planar coordinates of the equirectangular projection. 
* '''([[/EQUI2SPH/]])''' The reverse projection transforms equirectangular coordinates from the plane back onto the sphere. The formulae presume a [[w:en:figure of the Earth|spherical model]] 

=== Spherical - Longitude and Latitude - SPH ===
* Longitude &lt;math&gt;\lambda \in [ -180^\circ , +180^\circ ] &lt;/math&gt;
* Latitude  &lt;math&gt;\phi \in [ -90^\circ , +90^\circ ]  &lt;/math&gt;

=== Visualization ===
A perspective view of the Earth showing how latitude (&lt;math&gt;\phi&lt;/math&gt;) and longitude (&lt;math&gt;\lambda&lt;/math&gt;) are defined on a spherical model. The graticule spacing is 10 degrees.
[[File:latitude and longitude graticule on a sphere.svg|center|350px|A perspective view of the Earth showing how latitude and longitude]]

==== Longitude - SPH ====
Longitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Greenich Meridian as the [[w:en:Prime Meridian|Prime Meridian]] and is ranging to &lt;math&gt;+180^o&lt;/math&gt; eastward and &lt;math&gt;-180^o&lt;/math&gt; westward. The Greek letter &lt;math&gt;\lambda&lt;/math&gt; (lambda)&lt;ref&gt;{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|title=Coordinate Conversion|website=colorado.edu|access-date=14 March 2018|archive-url=https://web.archive.org/web/20090929121405/http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|archive-date=29 September 2009|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;&quot;&lt;math&gt;\lambda&lt;/math&gt; = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign).&quot;&lt;br /&gt;John P. Snyder, ''[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 Map Projections, A Working Manual] {{Webarchive|url=https://web.archive.org/web/20100701103721/http://pubs.er.usgs.gov/usgspubs/pp/pp1395 |date=2010-07-01 }}'', [[w:en:USGS|USGS]] Professional Paper 1395, page ix&lt;/ref&gt; is used to denote the location of a place on Earth east or west of the Prime Meridian

==== Latitude - SPH ====
Latitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Equator and is ranging to &lt;math&gt;+90^o&lt;/math&gt; towards the North Pole and &lt;math&gt;-90^o&lt;/math&gt; towards the South Pole. The Greek letter &lt;math&gt;\phi&lt;/math&gt; or &lt;math&gt;\varphi&lt;/math&gt; (phi) denotes that angle.

==== Mnemonic - Greek Letter - Phi====
There are two different notations of the greek letter &lt;math&gt; \phi &lt;/math&gt; and &lt;math&gt; \varphi &lt;/math&gt;. In this learning resource the notation &lt;math&gt; \phi &lt;/math&gt; is used to indicate that it denotes the angle at circle that intersects with the North Pole and the South Pole.

=== Definition of Spherical Variables ===
The projections are [[w:en:Function_(mathematics)|mathematical function/mappings]]. For definition of these projections the following variables are defined:
*&lt;math&gt;\lambda&lt;/math&gt; is the [[w:en:longitude|longitude]] of the location to project;
* &lt;math&gt;\phi&lt;/math&gt; is the [[w:en:latitude|latitude]] of the location to project;
* &lt;math&gt;\phi_1&lt;/math&gt; are the standard parallels (north and south of the equator) where the scale of the projection is true;
* &lt;math&gt;\phi_0&lt;/math&gt; is the central parallel of the map (e.g. &lt;math&gt;\phi_0 = 0^\circ &lt;/math&gt; equator);
* &lt;math&gt;\lambda_0&lt;/math&gt; is the central meridian of the map;
* &lt;math&gt;R&lt;/math&gt; is the radius of the globe.
Longitude and latitude variables are defined here in terms of radians.

=== Definition of Equirectangular Planar Variables - EQUI ===
* &lt;math&gt;x_e&lt;/math&gt; is the horizontal coordinate of the projected location on the map;
* &lt;math&gt;y_e&lt;/math&gt; is the vertical coordinate of the projected location on the map;

=== Forward Projection - Spherical to Planar - SPH2EQUI ===
&lt;math&gt;\begin{align}
x &amp;= R \cdot (\lambda - \lambda_0) \cdot \cos (\phi_1)\\
y &amp;= R \cdot (\phi - \phi_0)
\end{align}&lt;/math&gt;

=== Special Case - Forward Projection ===
The {{lang|fr|plate carrée}} ([[w:en:French language|French]], for ''flat square''),&lt;ref&gt;{{Cite web |title=Plate Carrée - a simple example |last=Farkas |first=Gábor |work=O’Reilly Online Learning |date= |access-date=31 December 2022 |url= https://www.oreilly.com/library/view/practical-gis/9781787123328/Text/b21938a9-09f7-46fa-b905-58a0a4ed7d8f.xhtml}}&lt;/ref&gt; is the special case where &lt;math&gt;\varphi_1&lt;/math&gt; is zero. This projection maps ''x'' to be the value of the longitude and ''y'' to be the value of the latitude,&lt;ref&gt;{{cite book |url=https://books.google.co.uk/books?id=-FbVI-2tSuYC&amp;pg=PA119 |p=119 |title=Geographic Information Systems and Science |author1=Paul A. Longley |author2=Michael F. Goodchild |author3=David J. Maguire |author4=David W. Rhind |publisher=John Wiley &amp; Sons |year=2005}}&lt;/ref&gt; and therefore is sometimes called the latitude/longitude or lat/lon(g) projection. 

When the &lt;math&gt;\phi_1&lt;/math&gt; is not zero, such as [[w:en:Marinus of Tyre|Marinus]]'s &lt;math&gt;\phi_1=36&lt;/math&gt;,&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;7, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; or [[w:en:Royal Scottish Geographical Society|Ronald Miller]]'s &lt;math&gt;\phi_1=(37.5, 43.5, 50.5)&lt;/math&gt;,&lt;ref&gt;{{cite web |title=Equidistant Cylindrical (Plate Carrée) |url=https://proj.org/operations/projections/eqc.html |website=PROJ coordinate transformation software library |access-date=25 August 2020}}&lt;/ref&gt; the projection can portray particular latitudes of interest at true scale.

=== Remarks - Ellipsoidal Model ===
While a projection with equally spaced parallels is possible for an '''ellipsoidal model''', it would no longer be equidistant because the distance between parallels on an ellipsoid is not constant. More complex formulae can be used to create an equidistant map whose parallels reflect the true spacing.

===Reverse - Planar to Spherical - EQUI2SPH ===
&lt;math&gt;\begin{align}
\lambda &amp;= \frac{x} {R \cdot \cos (\phi_1)} + \lambda_0\\
\phi &amp;= \frac{y} {R} + \phi_0
\end{align}&lt;/math&gt;

=== Alternative names ===
In spherical panorama viewers, usually:

* &lt;math&gt;\lambda&lt;/math&gt; is called &quot;yaw&quot;;&lt;ref&gt;{{cite web |title=Yaw - PanoTools.org Wiki |url=https://wiki.panotools.org/Yaw |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;
* &lt;math&gt;\phi&lt;/math&gt; is called &quot;pitch&quot;;&lt;ref&gt;{{cite web |title=Pitch - PanoTools.org Wiki |url=https://wiki.panotools.org/Pitch |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

where both are defined in degrees.

== Learning Activities ==
The following learning activities address [[w:en:Projective geometry|Projective Geometry]] from a standard snapshot taken with your camera onto an area on the sphere according to the equirectangular projection. Keep in mind to distinguish between the following projective converters:
*  '''(SPH2EQUI / EQUI2SPH):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x_e,y_e)&lt;/math&gt; coordinates of an equirectangular image
*  '''(SPH2IMG / IMG2SPH):''' &lt;math&gt;(x_e,y_e)&lt;/math&gt; equirectangular coordinates &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.
*  '''(EQUI2IMG / IMG2EQUI):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.

=== Angle of View ===
For the learning activities it is important to understand the 
* '''(HAV)''' Horizontal Angle of View in landscape format and 
* '''(VAV)''' Vertical Angle of View in landscape format. 

=== Learning Task - Angle of View ===
The HAV and VAV differ from camera to camera. Explain why the HAV and VAV are relevant for the calculation of IMG2EQUI projection. Use the following figure to explain the requirements for projection.

=== Visualization of Angle of View ===
The following figure depicts besides the Horizontal (HAV) and Vertical Angle of View (VAV) also the Diagonal Angle of View (DAV).

[[File:Angle of view.svg|350px|center|Horizontal, vertical and diagonal angle of view]]

=== Remark - IMG2EQUI Projection ===
The current projection type of the learning activities is the (IMG2EQUI) projection between the &lt;math&gt;(x,y)&lt;/math&gt; coordinates of a standard image  and the distorted &lt;math&gt;(x_e,y_e)&lt;/math&gt;  coordinates of an equirectangular image and vice verca.

=== HAV and VAV as camera specifc properties ===
Due to the fact that different cameras have different angles of view (HAV and VAV) the visible area of the standard image (taken with your smartphone) might vary. The following animation shows different horizontal angles of view (HAV) e.g. for taking a snapshot vertically upwards towards the sky or blue ceiling.

[[File:Horizontal angle of view.gif|350px|center|dynamic visualization of the Horizontal Angle of View]]

== Polar Region - Distortion ==
The projection creates especially at the &quot;North Pole&quot; and the &quot;South Pole&quot; the [[Maps and Distortion|heaviest distortion in comparison to distances measured]] on the surface of the sphere. For panoramic views the distortion is just a matter of storage of the spheric pixel information in a rectangular format. On the image of a market place you will see that the panoramic viewes transfer the rectangular images into a natural view where you can look around and explore the location from different angles and with multiple equirectangular images from many locations (see [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html river Rhine example Cologne]).

=== IMG2EQUI Projection - Polar Regions ===
In this learning step we take pictures with a standard camera or smartphone and want to calculate the ''(Latitude,Longitude)'' coordinates of the sphere from ''(x,y)'' coordinates of standard image created with your smartphone or camera.
In the first learning step we consider the sphere at the polar regions. These regions are the most distorted areas in the equirectangular projections.

=== IMG2SPH Projection - Take Snapshots for Learning Task ===
* Take your mobile phone and take two snapshots vertical down and vertical up from your floor and from the ceiling. Select an position in your room where the top and the floor has some visible elements (e.g. a lamp, wooden decorative elements,  ...). Alternatively you can create the two images outside with a cloudy sky and objects lying on the ground. 
* select a center of a circle and a radius that fits into both images (maximize the radius of the circle,
* in this module we will project these two images to the polar regions of the sphere within the rectangular coordinate system of the image.

==== Initial State - Image Sky ====
The following image shows the initial state taking a picture from the sky. The camera is located in the center of the red sphere and takes an image of the blue sky or the ceiling in a room. The ceiling is visualized as a blue plane. The learning task is to calculate the angles for the corresponding point on the sphere. 
 
[[File:Circle plane equirectangular.png|350px|Equirectangular Projection - Sky Image - Polar Projection]]

==== Side view with polar image ====
The side view can be used to derive the calculation of the green angle with &lt;math&gt;\alpha:=\arctan(x)&lt;/math&gt;. The relevant angle for the equirectangular projection of the polar region is marked red. 

[[File:Equirectangular polar side view.gif|350px|Equirectangular projection side view - relevant angle is the red angle]]

==== Used Part of Snapshot with a Camera ====
The first image is snapshot with a camera taking a picture vertically upwards to the ceiling (into the sky). The red circle is the projected area from the image (sky/ceiling) onto a part of the equirectangular image visualized in the next section of the learning resource.

[[File:Equirectangular polar image.svg|350px|Equirectangular source image with circle to be projected]]

==== Equirectangular Projection of Circle - Rectangle ====
Explain, why the circle in the source image of the polar region (North Pole) is a rectangle after projection in the equirectangular image.

[[File:Equirectangular polar image projection.svg|350px|Equirectangular source image with circle to be projected]]

==== Learning Task - Function for Coordinate Transformation IMG2EQUI ====
We denote by &lt;math&gt;P_{IMG}(x,y)&lt;/math&gt; the color information of a pixel in the source image (IMG) and with &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt;  the color information of the pixel in the image in the destination format (EQUI). Define the function &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt; by calculating the corresponding &lt;math&gt;(x,y)&lt;/math&gt; coordinates in the IMG source image. This calculation is used to set pixels in the destination format EQUI by setting  
:&lt;math display=&quot;block&quot;&gt;P_{EQUI}(x_e,y_e):=P_{IMG}(x,y).&lt;/math&gt;
'''Remark:''' You need basic knowledge in [[trigonometry]] to perform this learning task.

==== Coordinate System of Graphics ====
[[File:Graphics coordinate system.svg|thumb|Coordinate system in Graphics]]
The coordinate system of an image has a different orientation in y-axis. This is shown in the diagram.
* &lt;math&gt;x_{max}&lt;/math&gt; is the maximal value of the on the x-axis of the image
* &lt;math&gt;y_{max}&lt;/math&gt; is the maximal value of the on the y-axis  the of the image
* The origin of the coordinate system is on the top left. 
Keep this in mind, when you use the coordinate system for your experiments with projections (see also equirectangular projection.

==== Rectangular to Sphere - Projection ====

[[File:Equirectangular ceiling floor.jpg|thumb|Equirectangular projection with marked ceiling and floor]]
Now we use a the standard rectangular image on the right as in input for the equirectangular projection on the sphere and we view the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image in panoramic preview on the sphere]. What can be observed, if you analyze the projected area of red [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg ceiling and a marked green rectangle on the floor].
* Drag the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image upwards to the ceiling] 
* and [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg downwards to floor].

==== Screenshots - Projection on Sphere of the standard Image ====
{| class=&quot;wikitable&quot;
|+ Screenshot of projection
|-
! Ceiling/Sky !! Floor
|-
| [[File:Equirectangular screenshot ceiling.png|320px|Screenshot ceiling/sky of standard image projected on a sphere with equirectangular projection]] || [[File:Equirectangular screenshot floor.png|320px|Screenshot floor of standard image projected on a sphere with equirectangular projection]]
|}

==== Learning Task - Projection of Sky in Graphics Coordinate System ====

[[File:Equirectangular coordinate system sky.svg|thumb|Equirectangular coordinate system sky / ceiling]]

Graphics have an own coordinate system to display points in pixel graphics or geometric objects like lines, polygons and circles in the coordinate system. The diagram shows the coordinate system of the image. Keep in mind that the coordinate of the y-axis has a different orientation than the y-axis in the standard 2D [[w:de:Cartesian coordinate system|Cartesian coordinate system]]. This information is relevant if you experiment with equirectangular projections in [[Projective Geometry Playground]].

Calculate the &lt;math&gt;y_{sky}&lt;/math&gt; for a specific angle of view of your camera.

==== Learning Task - Calculation ====
[[File:Horizontal angle of view.jpg|thumb|Fullscreen image of Door for HAV/VAV calc
[[File:Horizontal angle of view meaure distance.jpg|thumb|Measure distance from camera position to door ]]
ulation]]

Calculate the equirectangular projection for any &lt;math&gt;(x,y)&lt;/math&gt; point in the red circle of the source image IMG into the destination format EQUI with:
* '''LibreOffice - HAV/VAV:''' Take image of a door and calculate the horizontal and vertical angle of view (HAV and VAV)
* '''LibreOffice - Coordinates:''' Create a Spreadsheet document for the calculation of coordinates from a given coordinate in the equirectangular image the corresponding coordinates in the source image of your camera.
* [[Projective Geometry Playground|Javascript and HTML canvas]] - (see [[Projective Geometry Playground]]),
* [[w:en:GNU Octave|Octave]] with [https://gnu-octave.github.io/packages/image/ image-package]&lt;ref&gt; Carnë Draug,
Hartmut Gimpel, Avinoam Kalma (2022) Image Package Octave URL: https://gnu-octave.github.io/packages/image/ (March, 28th, 2024)&lt;/ref&gt;, or 
* Python with [https://github.com/python-pillow/Pillow Image Processing Library pillow]  by Jeffrey A. Clark   
Select an implementation of your choice. LibreOffice has the minimal requirements on programming skill but only coordinate transformation can be performed without a visual output.

=== Learning Task - Sky - North Pole ===
[[File:Sky image for equirectangular projection.jpg|thumb|Sky - equirectangular projection - north pole]]
A sky image can be used to project a circular area in the source image to the rectangular part at the top of the generated equirectangular image. 

Import the equirectangular projection in Aframe to preview the result.

==== Learning Task - Floor - South Pole ====
[[File:Floor sand for equirectangular.jpg|thumb|Floor Image - sand as demo input for equirectangular projection]]Transfer the lesson learned from north pole to the south pole and project the beach image as floor to the south pole. 
* What are differences and similarities between both projections?

== Final Result ==
[[File:Aldara parks.jpg|thumb|[https://niebert.github.io/HuginSample/Aldara_parks.html Equirectangular Image from Wikiversity used for Aframe 360 Degree Image] (see [[3D Modelling/Create 3D Models/Hugin|Hugin]])]]
{{PanoViewer|Frary Dining Hall 360-degree view.jpg|Dining Hall 360-degree view - with PanoViewer Template}}
* [https://niebert.github.io/HuginSample/ Final Results]  would can be previewed in Aframe (see [https://www.github.com/HuginSample HuginSample Files on Github]) or with the Wiki,
* [https://niebert.github.com/HuginSample/Aldara_parks.html Aldara Parks 360 Degree Image in Aframe] with an already uploaded equirectangular image in WikiMedia Commons by U.Bardins
* [https://panoviewer.toolforge.org/#Aldara_parks.jpg PanoViewer 360 degree view of Aldara Parks image]]

==See also==
* [[Projective Geometry Playground]]
* [[GeoGebra/Perspective Drawing on Mirror|Perspective Drawing on Mirror]]
* [[3D Modelling]]
* [[w:en:Cartography|Cartography]]
* [[w:en:Cassini projection|Cassini projection]]
* [[w:en:Gall–Peters projection|Gall–Peters projection]] with resolution regarding the use of rectangular world maps
* [[w:en:List of map projections|List of map projections]]
* [[w:en:Mercator projection|Mercator projection]]
* [[w:en:360 video projection|360 video projection]]
* [[3D_Modelling/Examples/Panorama_360|3D Modelling - 360 degree panorama]]
* [[Geometry]]
* [[w:en:Projective_geometry|Projective Geometry]]
* [[Portal:Mathematics]]
* [[w:en:GNU Octave|Octave]]
* [[Geogebra]]
* [[b:en:Javascript|Wikibook: Javascript]]
* [[Portal:Mathematics]]

==References==
{{Reflist}}

==External links==
* [https://visibleearth.nasa.gov/view.php?id=57730 Global MODIS based satellite map] The blue marble: land surface, ocean color, and sea ice.
* [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net.
* [http://wiki.panotools.org/Equirectangular Panoramic Equirectangular Projection], PanoTools wiki.
* [https://proj4.org/operations/projections/eqc.html Equidistant Cylindrical (Plate Carrée) in proj4]

== Page Information ==
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=== Wiki2Reveal ===
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=== Wikipedia2Wikiversiy===
This page was based on the following [https://en.wikipedia.org/wiki/Equirectangular%20projection wikipedia-source page]:
* [https://en.wikipedia.org/wiki/Equirectangular%20projection Equirectangular projection] https://en.wikipedia.org/wiki/Equirectangular%20projection
* Datum: 1/9/2023
* [https://niebert.github.io/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.io/Wikipedia2Wikiversity


[[Category:Map projections]]
[[Category:Equidistant projections]]
[[Category:Cylindrical projections]]
[[Category:Wiki2Reveal]]</text>
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      <text bytes="26763" xml:space="preserve">== Wiki2Reveal ==
This learning resource can be used as [[Wiki2Reveal]] slides in mathematics courses as introduction to [[w:en:Projective geometry|projective geometry]].

* Start '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=Geometry&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=Geometry Wiki2Reveal]''' [[File:Wiki2Reveal Logo.png|35px]]
  

== Introduction == 
In this learning resource the generation of an equirectangular projection is the objective. The learning resource is use-case driven, [https://niebert.github.io/HuginSample/ AFrame Example Durlach] is the first use-case of equirectangular projection. Look around by dragging the direction of view with the mouse with left mouse button pressed.
[[File:Hugin result in aframe.png|center|300px|Equirectangular image used in spheric image in a browser ]]
&lt;center&gt;
[https://niebert.github.io/HuginSample/ Preview of the equirectangular projection in AFrame] - Drag the preview of the equirectangular image with your mouse button pressed.
&lt;/center&gt;

=== Example of the equirectangular image ===  
Preview the [https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg equirectangular JPG-image] and explore the distortion of the image at the top and the bottom of the rectangular JPEG image. 

=== Navigation with multiple equirectangular images ===
The [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html AFrame Navigation example from the river Rhine in Germany] allows to jump from one location at the river rhine to another location at to the equirectangular preview.
 
[[File:river_rhine_spheric_preview.png|center|300px| - Use case of equirectangular projection  - River Rhine example with multiple locations]]

== Subtopics ==
* [[/Maps and Distortion/]]

== Use-Case ==
From a set of standard images of the camera covering a full the from center point of view. In general you cover 360 degree circle with rectangular standard images and you take image with camera for the sky and the floor (preferred without seeing the tripod that might be used for the other images). These set of images are aggregate in one equirectangular image representing a [https://niebert.github.io/HuginSample/ full spheric panorama image], that be be viewed e.g. in [https://aframe.io/sky Aframe] or other panoramic OpenSource viewes, that support equirectangular images. 

=== EQUI2SPH Projection ===
The underlying type of projection is an equirectangular projection EQUI2SPH, that is used e.g in geographical context, where a sphere is projected to rectangular plane on the map. First of all we explore the use-case in [[3D Modelling]] about the spherical use of an spherical panorama image.

== Origin of Terminology and History ==
The ''equirectangular projection'' (also called the ''equidistant cylindrical projection''), and which includes the special case of the ''plate carrée projection'' (also called the ''geographic projection'', ''lat/lon projection'', or ''plane chart''), is a simple [[w:en:map projection|map projection]] attributed to [[w:en:Marinus of Tyre|Marinus of Tyre]], who [[w:en:Ptolemy|Ptolemy]] claims invented the projection about AD 100.&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;5–8, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; The projection maps [[w:en:meridian (geography)|meridians]] to vertical straight lines of constant spacing (for meridional intervals of constant spacing), and [[w:en:circle of latitude|circles of latitude]] to horizontal straight lines of constant spacing (for constant intervals of [[w:en:circle of latitude|parallels]]). The projection is neither [[w:en:equal-area map|equal area]] nor [[w:en:conformal map projection|conformal]]. 

=== Implications of Distortion for Navigation ===
Because of the distortions introduced by this projection, it has little use in [[w:en:navigation|navigation]] or [[w:en:cadastral|cadastral]] mapping and finds its main use in [[w:en:thematic map|thematic mapping]]. 

=== Application in global Raster Datasets ===
In particular, the plate carrée has become a standard for global [[w:en:geographic information system|raster datasets]], such as [[w:en:Celestia|Celestia]], [[w:en:NASA World Wind|NASA World Wind]], the [[w:en:USGS|USGS]] [[w:en:Astrogeology Research Program|Astrogeology Research Program]], and [[w:en:Natural Earth|Natural Earth]], because of the particularly simple relationship between the position of an [[w:en:pixel|image pixel]] on the map and its corresponding geographic location on Earth or other spherical solar system bodies. 

=== Application in panoramic photography ===
In addition it is frequently used in panoramic photography to represent a spherical panoramic image.&lt;ref&gt;{{cite web |title=Equirectangular Projection - PanoTools.org Wiki |url=https://wiki.panotools.org/Equirectangular_Projection |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

== Definition - Equirectangular Projection ==
* '''([[/SPH2EQUI/]])''' The forward projection transforms spherical coordinates into planar coordinates of the equirectangular projection. 
* '''([[/EQUI2SPH/]])''' The reverse projection transforms equirectangular coordinates from the plane back onto the sphere. The formulae presume a [[w:en:figure of the Earth|spherical model]] 

=== Spherical - Longitude and Latitude - SPH ===
* Longitude &lt;math&gt;\lambda \in [ -180^\circ , +180^\circ ] &lt;/math&gt;
* Latitude  &lt;math&gt;\phi \in [ -90^\circ , +90^\circ ]  &lt;/math&gt;

=== Visualization ===
A perspective view of the Earth showing how latitude (&lt;math&gt;\phi&lt;/math&gt;) and longitude (&lt;math&gt;\lambda&lt;/math&gt;) are defined on a spherical model. The graticule spacing is 10 degrees.
[[File:latitude and longitude graticule on a sphere.svg|center|350px|A perspective view of the Earth showing how latitude and longitude]]

==== Longitude - SPH ====
Longitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Greenich Meridian as the [[w:en:Prime Meridian|Prime Meridian]] and is ranging to &lt;math&gt;+180^o&lt;/math&gt; eastward and &lt;math&gt;-180^o&lt;/math&gt; westward. The Greek letter &lt;math&gt;\lambda&lt;/math&gt; (lambda)&lt;ref&gt;{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|title=Coordinate Conversion|website=colorado.edu|access-date=14 March 2018|archive-url=https://web.archive.org/web/20090929121405/http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|archive-date=29 September 2009|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;&quot;&lt;math&gt;\lambda&lt;/math&gt; = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign).&quot;&lt;br /&gt;John P. Snyder, ''[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 Map Projections, A Working Manual] {{Webarchive|url=https://web.archive.org/web/20100701103721/http://pubs.er.usgs.gov/usgspubs/pp/pp1395 |date=2010-07-01 }}'', [[w:en:USGS|USGS]] Professional Paper 1395, page ix&lt;/ref&gt; is used to denote the location of a place on Earth east or west of the Prime Meridian

==== Latitude - SPH ====
Latitude is given as an [[w:en:angle|angular measurement]] referring to  the 0° Equator and is ranging to &lt;math&gt;+90^o&lt;/math&gt; towards the North Pole and &lt;math&gt;-90^o&lt;/math&gt; towards the South Pole. The Greek letter &lt;math&gt;\phi&lt;/math&gt; or &lt;math&gt;\varphi&lt;/math&gt; (phi) denotes that angle.

==== Mnemonic - Greek Letter - Phi====
There are two different notations of the greek letter &lt;math&gt; \phi &lt;/math&gt; and &lt;math&gt; \varphi &lt;/math&gt;. In this learning resource the notation &lt;math&gt; \phi &lt;/math&gt; is used to indicate that it denotes the angle at circle that intersects with the North Pole and the South Pole.

=== Definition of Spherical Variables ===
The projections are [[w:en:Function_(mathematics)|mathematical function/mappings]]. For definition of these projections the following variables are defined:
*&lt;math&gt;\lambda&lt;/math&gt; is the [[w:en:longitude|longitude]] of the location to project;
* &lt;math&gt;\phi&lt;/math&gt; is the [[w:en:latitude|latitude]] of the location to project;
* &lt;math&gt;\phi_1&lt;/math&gt; are the standard parallels (north and south of the equator) where the scale of the projection is true;
* &lt;math&gt;\phi_0&lt;/math&gt; is the central parallel of the map (e.g. &lt;math&gt;\phi_0 = 0^\circ &lt;/math&gt; equator);
* &lt;math&gt;\lambda_0&lt;/math&gt; is the central meridian of the map;
* &lt;math&gt;R&lt;/math&gt; is the radius of the globe.
Longitude and latitude variables are defined here in terms of radians.

=== Definition of Equirectangular Planar Variables - EQUI ===
* &lt;math&gt;x_e&lt;/math&gt; is the horizontal coordinate of the projected location on the map;
* &lt;math&gt;y_e&lt;/math&gt; is the vertical coordinate of the projected location on the map;

=== Forward Projection - Spherical to Planar - SPH2EQUI ===
&lt;math&gt;\begin{align}
x &amp;= R \cdot (\lambda - \lambda_0) \cdot \cos (\phi_1)\\
y &amp;= R \cdot (\phi - \phi_0)
\end{align}&lt;/math&gt;

=== Special Case - Forward Projection ===
The {{lang|fr|plate carrée}} ([[w:en:French language|French]], for ''flat square''),&lt;ref&gt;{{Cite web |title=Plate Carrée - a simple example |last=Farkas |first=Gábor |work=O’Reilly Online Learning |date= |access-date=31 December 2022 |url= https://www.oreilly.com/library/view/practical-gis/9781787123328/Text/b21938a9-09f7-46fa-b905-58a0a4ed7d8f.xhtml}}&lt;/ref&gt; is the special case where &lt;math&gt;\varphi_1&lt;/math&gt; is zero. This projection maps ''x'' to be the value of the longitude and ''y'' to be the value of the latitude,&lt;ref&gt;{{cite book |url=https://books.google.co.uk/books?id=-FbVI-2tSuYC&amp;pg=PA119 |p=119 |title=Geographic Information Systems and Science |author1=Paul A. Longley |author2=Michael F. Goodchild |author3=David J. Maguire |author4=David W. Rhind |publisher=John Wiley &amp; Sons |year=2005}}&lt;/ref&gt; and therefore is sometimes called the latitude/longitude or lat/lon(g) projection. 

When the &lt;math&gt;\phi_1&lt;/math&gt; is not zero, such as [[w:en:Marinus of Tyre|Marinus]]'s &lt;math&gt;\phi_1=36&lt;/math&gt;,&lt;ref&gt;''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&amp;nbsp;7, {{ISBN|0-226-76747-7}}.&lt;/ref&gt; or [[w:en:Royal Scottish Geographical Society|Ronald Miller]]'s &lt;math&gt;\phi_1=(37.5, 43.5, 50.5)&lt;/math&gt;,&lt;ref&gt;{{cite web |title=Equidistant Cylindrical (Plate Carrée) |url=https://proj.org/operations/projections/eqc.html |website=PROJ coordinate transformation software library |access-date=25 August 2020}}&lt;/ref&gt; the projection can portray particular latitudes of interest at true scale.

=== Remarks - Ellipsoidal Model ===
While a projection with equally spaced parallels is possible for an '''ellipsoidal model''', it would no longer be equidistant because the distance between parallels on an ellipsoid is not constant. More complex formulae can be used to create an equidistant map whose parallels reflect the true spacing.

===Reverse - Planar to Spherical - EQUI2SPH ===
&lt;math&gt;\begin{align}
\lambda &amp;= \frac{x} {R \cdot \cos (\phi_1)} + \lambda_0\\
\phi &amp;= \frac{y} {R} + \phi_0
\end{align}&lt;/math&gt;

=== Alternative names ===
In spherical panorama viewers, usually:

* &lt;math&gt;\lambda&lt;/math&gt; is called &quot;yaw&quot;;&lt;ref&gt;{{cite web |title=Yaw - PanoTools.org Wiki |url=https://wiki.panotools.org/Yaw |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;
* &lt;math&gt;\phi&lt;/math&gt; is called &quot;pitch&quot;;&lt;ref&gt;{{cite web |title=Pitch - PanoTools.org Wiki |url=https://wiki.panotools.org/Pitch |access-date=2021-05-04 |website=wiki.panotools.org}}&lt;/ref&gt;

where both are defined in degrees.

== Learning Activities ==
The following learning activities address [[w:en:Projective geometry|Projective Geometry]] from a standard snapshot taken with your camera onto an area on the sphere according to the equirectangular projection. Keep in mind to distinguish between the following projective converters:
*  '''(SPH2EQUI / EQUI2SPH):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x_e,y_e)&lt;/math&gt; coordinates of an equirectangular image
*  '''(SPH2IMG / IMG2SPH):''' &lt;math&gt;(x_e,y_e)&lt;/math&gt; equirectangular coordinates &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.
*  '''(EQUI2IMG / IMG2EQUI):''' ''(Latitude,Longitude)'' coordinates sphere &lt;math&gt;\Longleftrightarrow &lt;/math&gt; &lt;math&gt;(x,y)&lt;/math&gt; coordinates standard image of your smartphone or camera.

=== Angle of View ===
For the learning activities it is important to understand the 
* '''(HAV)''' Horizontal Angle of View in landscape format and 
* '''(VAV)''' Vertical Angle of View in landscape format. 

=== Learning Task - Angle of View ===
The HAV and VAV differ from camera to camera. Explain why the HAV and VAV are relevant for the calculation of IMG2EQUI projection. Use the following figure to explain the requirements for projection.

=== Visualization of Angle of View ===
The following figure depicts besides the Horizontal (HAV) and Vertical Angle of View (VAV) also the Diagonal Angle of View (DAV).

[[File:Angle of view.svg|350px|center|Horizontal, vertical and diagonal angle of view]]

=== Remark - IMG2EQUI Projection ===
The current projection type of the learning activities is the (IMG2EQUI) projection between the &lt;math&gt;(x,y)&lt;/math&gt; coordinates of a standard image  and the distorted &lt;math&gt;(x_e,y_e)&lt;/math&gt;  coordinates of an equirectangular image and vice verca.

=== HAV and VAV as camera specifc properties ===
Due to the fact that different cameras have different angles of view (HAV and VAV) the visible area of the standard image (taken with your smartphone) might vary. The following animation shows different horizontal angles of view (HAV) e.g. for taking a snapshot vertically upwards towards the sky or blue ceiling.

[[File:Horizontal angle of view.gif|350px|center|dynamic visualization of the Horizontal Angle of View]]

== Polar Region - Distortion ==
The projection creates especially at the &quot;North Pole&quot; and the &quot;South Pole&quot; the [[/Maps and Distortion/|heaviest distortion in comparison to distances measured]] on the surface of the sphere. For panoramic views the distortion is just a matter of storage of the spheric pixel information in a rectangular format. On the image of a market place you will see that the panoramic viewes transfer the rectangular images into a natural view where you can look around and explore the location from different angles and with multiple equirectangular images from many locations (see [https://niebert.github.io/aframe360navigation/rhein3_rodenkirchen.html river Rhine example Cologne]).

=== IMG2EQUI Projection - Polar Regions ===
In this learning step we take pictures with a standard camera or smartphone and want to calculate the ''(Latitude,Longitude)'' coordinates of the sphere from ''(x,y)'' coordinates of standard image created with your smartphone or camera.
In the first learning step we consider the sphere at the polar regions. These regions are the most distorted areas in the equirectangular projections.

=== IMG2SPH Projection - Take Snapshots for Learning Task ===
* Take your mobile phone and take two snapshots vertical down and vertical up from your floor and from the ceiling. Select an position in your room where the top and the floor has some visible elements (e.g. a lamp, wooden decorative elements,  ...). Alternatively you can create the two images outside with a cloudy sky and objects lying on the ground. 
* select a center of a circle and a radius that fits into both images (maximize the radius of the circle,
* in this module we will project these two images to the polar regions of the sphere within the rectangular coordinate system of the image.

==== Initial State - Image Sky ====
The following image shows the initial state taking a picture from the sky. The camera is located in the center of the red sphere and takes an image of the blue sky or the ceiling in a room. The ceiling is visualized as a blue plane. The learning task is to calculate the angles for the corresponding point on the sphere. 
 
[[File:Circle plane equirectangular.png|350px|Equirectangular Projection - Sky Image - Polar Projection]]

==== Side view with polar image ====
The side view can be used to derive the calculation of the green angle with &lt;math&gt;\alpha:=\arctan(x)&lt;/math&gt;. The relevant angle for the equirectangular projection of the polar region is marked red. 

[[File:Equirectangular polar side view.gif|350px|Equirectangular projection side view - relevant angle is the red angle]]

==== Used Part of Snapshot with a Camera ====
The first image is snapshot with a camera taking a picture vertically upwards to the ceiling (into the sky). The red circle is the projected area from the image (sky/ceiling) onto a part of the equirectangular image visualized in the next section of the learning resource.

[[File:Equirectangular polar image.svg|350px|Equirectangular source image with circle to be projected]]

==== Equirectangular Projection of Circle - Rectangle ====
Explain, why the circle in the source image of the polar region (North Pole) is a rectangle after projection in the equirectangular image.

[[File:Equirectangular polar image projection.svg|350px|Equirectangular source image with circle to be projected]]

==== Learning Task - Function for Coordinate Transformation IMG2EQUI ====
We denote by &lt;math&gt;P_{IMG}(x,y)&lt;/math&gt; the color information of a pixel in the source image (IMG) and with &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt;  the color information of the pixel in the image in the destination format (EQUI). Define the function &lt;math&gt;P_{EQUI}(x_e,y_e)&lt;/math&gt; by calculating the corresponding &lt;math&gt;(x,y)&lt;/math&gt; coordinates in the IMG source image. This calculation is used to set pixels in the destination format EQUI by setting  
:&lt;math display=&quot;block&quot;&gt;P_{EQUI}(x_e,y_e):=P_{IMG}(x,y).&lt;/math&gt;
'''Remark:''' You need basic knowledge in [[trigonometry]] to perform this learning task.

==== Coordinate System of Graphics ====
[[File:Graphics coordinate system.svg|thumb|Coordinate system in Graphics]]
The coordinate system of an image has a different orientation in y-axis. This is shown in the diagram.
* &lt;math&gt;x_{max}&lt;/math&gt; is the maximal value of the on the x-axis of the image
* &lt;math&gt;y_{max}&lt;/math&gt; is the maximal value of the on the y-axis  the of the image
* The origin of the coordinate system is on the top left. 
Keep this in mind, when you use the coordinate system for your experiments with projections (see also equirectangular projection.

==== Rectangular to Sphere - Projection ====

[[File:Equirectangular ceiling floor.jpg|thumb|Equirectangular projection with marked ceiling and floor]]
Now we use a the standard rectangular image on the right as in input for the equirectangular projection on the sphere and we view the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image in panoramic preview on the sphere]. What can be observed, if you analyze the projected area of red [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg ceiling and a marked green rectangle on the floor].
* Drag the [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg image upwards to the ceiling] 
* and [https://panoviewer.toolforge.org/#Equirectangular_ceiling_floor.jpg downwards to floor].

==== Screenshots - Projection on Sphere of the standard Image ====
{| class=&quot;wikitable&quot;
|+ Screenshot of projection
|-
! Ceiling/Sky !! Floor
|-
| [[File:Equirectangular screenshot ceiling.png|320px|Screenshot ceiling/sky of standard image projected on a sphere with equirectangular projection]] || [[File:Equirectangular screenshot floor.png|320px|Screenshot floor of standard image projected on a sphere with equirectangular projection]]
|}

==== Learning Task - Projection of Sky in Graphics Coordinate System ====

[[File:Equirectangular coordinate system sky.svg|thumb|Equirectangular coordinate system sky / ceiling]]

Graphics have an own coordinate system to display points in pixel graphics or geometric objects like lines, polygons and circles in the coordinate system. The diagram shows the coordinate system of the image. Keep in mind that the coordinate of the y-axis has a different orientation than the y-axis in the standard 2D [[w:de:Cartesian coordinate system|Cartesian coordinate system]]. This information is relevant if you experiment with equirectangular projections in [[Projective Geometry Playground]].

Calculate the &lt;math&gt;y_{sky}&lt;/math&gt; for a specific angle of view of your camera.

==== Learning Task - Calculation ====
[[File:Horizontal angle of view.jpg|thumb|Fullscreen image of Door for HAV/VAV calc
[[File:Horizontal angle of view meaure distance.jpg|thumb|Measure distance from camera position to door ]]
ulation]]

Calculate the equirectangular projection for any &lt;math&gt;(x,y)&lt;/math&gt; point in the red circle of the source image IMG into the destination format EQUI with:
* '''LibreOffice - HAV/VAV:''' Take image of a door and calculate the horizontal and vertical angle of view (HAV and VAV)
* '''LibreOffice - Coordinates:''' Create a Spreadsheet document for the calculation of coordinates from a given coordinate in the equirectangular image the corresponding coordinates in the source image of your camera.
* [[Projective Geometry Playground|Javascript and HTML canvas]] - (see [[Projective Geometry Playground]]),
* [[w:en:GNU Octave|Octave]] with [https://gnu-octave.github.io/packages/image/ image-package]&lt;ref&gt; Carnë Draug,
Hartmut Gimpel, Avinoam Kalma (2022) Image Package Octave URL: https://gnu-octave.github.io/packages/image/ (March, 28th, 2024)&lt;/ref&gt;, or 
* Python with [https://github.com/python-pillow/Pillow Image Processing Library pillow]  by Jeffrey A. Clark   
Select an implementation of your choice. LibreOffice has the minimal requirements on programming skill but only coordinate transformation can be performed without a visual output.

=== Learning Task - Sky - North Pole ===
[[File:Sky image for equirectangular projection.jpg|thumb|Sky - equirectangular projection - north pole]]
A sky image can be used to project a circular area in the source image to the rectangular part at the top of the generated equirectangular image. 

Import the equirectangular projection in Aframe to preview the result.

==== Learning Task - Floor - South Pole ====
[[File:Floor sand for equirectangular.jpg|thumb|Floor Image - sand as demo input for equirectangular projection]]Transfer the lesson learned from north pole to the south pole and project the beach image as floor to the south pole. 
* What are differences and similarities between both projections?

== Final Result ==
[[File:Aldara parks.jpg|thumb|[https://niebert.github.io/HuginSample/Aldara_parks.html Equirectangular Image from Wikiversity used for Aframe 360 Degree Image] (see [[3D Modelling/Create 3D Models/Hugin|Hugin]])]]
{{PanoViewer|Frary Dining Hall 360-degree view.jpg|Dining Hall 360-degree view - with PanoViewer Template}}
* [https://niebert.github.io/HuginSample/ Final Results]  would can be previewed in Aframe (see [https://www.github.com/HuginSample HuginSample Files on Github]) or with the Wiki,
* [https://niebert.github.com/HuginSample/Aldara_parks.html Aldara Parks 360 Degree Image in Aframe] with an already uploaded equirectangular image in WikiMedia Commons by U.Bardins
* [https://panoviewer.toolforge.org/#Aldara_parks.jpg PanoViewer 360 degree view of Aldara Parks image]]

==See also==
* [[Projective Geometry Playground]]
* [[GeoGebra/Perspective Drawing on Mirror|Perspective Drawing on Mirror]]
* [[3D Modelling]]
* [[w:en:Cartography|Cartography]]
* [[w:en:Cassini projection|Cassini projection]]
* [[w:en:Gall–Peters projection|Gall–Peters projection]] with resolution regarding the use of rectangular world maps
* [[w:en:List of map projections|List of map projections]]
* [[w:en:Mercator projection|Mercator projection]]
* [[w:en:360 video projection|360 video projection]]
* [[3D_Modelling/Examples/Panorama_360|3D Modelling - 360 degree panorama]]
* [[Geometry]]
* [[w:en:Projective_geometry|Projective Geometry]]
* [[Portal:Mathematics]]
* [[w:en:GNU Octave|Octave]]
* [[Geogebra]]
* [[b:en:Javascript|Wikibook: Javascript]]
* [[Portal:Mathematics]]

==References==
{{Reflist}}

==External links==
* [https://visibleearth.nasa.gov/view.php?id=57730 Global MODIS based satellite map] The blue marble: land surface, ocean color, and sea ice.
* [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net.
* [http://wiki.panotools.org/Equirectangular Panoramic Equirectangular Projection], PanoTools wiki.
* [https://proj4.org/operations/projections/eqc.html Equidistant Cylindrical (Plate Carrée) in proj4]

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** [https://en.wikipedia.org/wiki/Equirectangular%20projection https://en.wikiversity.org/wiki/Equirectangular%20projection]
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* [https://en.wikiversity.org/wiki/Equirectangular%20projection This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
* Source: Wikiversity  https://en.wikiversity.org/wiki/Equirectangular%20projection
* see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Equirectangular%20projection&amp;author=3D%20Modelling&amp;language=en&amp;audioslide=yes&amp;shorttitle=Equirectangular%20projection&amp;coursetitle=3D%20Modelling Wiki2Reveal].
&lt;!-- * Next contents of the course are [[]] --&gt;;

=== Wikipedia2Wikiversiy===
This page was based on the following [https://en.wikipedia.org/wiki/Equirectangular%20projection wikipedia-source page]:
* [https://en.wikipedia.org/wiki/Equirectangular%20projection Equirectangular projection] https://en.wikipedia.org/wiki/Equirectangular%20projection
* Datum: 1/9/2023
* [https://niebert.github.io/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.io/Wikipedia2Wikiversity


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    <title>Which is the best religion to follow?</title>
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There are hundreds of different religions in the world. Leaving aside the case of people who do not wish to follow any religion, is there an objective way to measure which of them is better? If yes, which one is the best for humans? For a debate about the existence of God, see [[Does God exist?]]

=== Limitations ===
* This page only covers the top 4 religions in the world (Hinduism, Christianity, Buddhism and Islam). There is no way it could cover all religions.
* The word &quot;best&quot; refers to the word &quot;good&quot;, which arguably is ambiguous or vague. It is up to the arguments to interpret &quot;good&quot; and &quot;best&quot; in reference to some chosen standard/criterion of good.
* Even given a fixed standard of good, a demonstration that a particular religion is better than all other religions seems nearly impossible. Therefore, the debate will probably have to be constrained to identifying good and bad aspects of various religions, with respect to various standards of ''good''.

== Hinduism ==

=== Pro ===

* {{Argument for}} [[Hinduism]] believes in unity in diversity. It never feels bad to welcome agnostics and atheists to practicing their own belief.
* {{Argument for}} Hinduism accepts change as fundamental nature of the universe. Thus, it goes well with changing societal norms.
* {{Argument for}} Science and technology has evolved and was learnt from Hindu scriptures
* {{Argument for}} Protects rights of everyone, men, women, children and even animals. Everyone and every being is worshipped, respected and prayed to.
** {{Objection}} How can we worshipe everyone the definition of god is the one who posses infinite powers and no one should be powerful than the god The good should be one to whom should we always seek help
** {{Argument for}} God is omnipotent and supreme. He has infinite power beyond comprehension. He manifests himself in all beings, and all spirituality, every soul is connected to him. That is the reason every being is respected and worshipped. Not only does it follow the rules of spirituality, but basic humanity also to care for every being.
&lt;ref&gt;{{Cite web|url=https://vishnueverywhere.tumblr.com/|title=Vishnu Everywhere|last=|website=Vishnu is the master of-and beyond-the past, present and future, one who supports, sustains and governs the Universe and originates and develops all elements within.|access-date=2024-04-15}}&lt;/ref&gt; Check - https://vishnueverywhere.tumblr.com/
* {{Argument for}} There are no strict rules. One should abide by Dharma or also called Righteousness (duty). God is everywhere and in every being.
* {{Argument for}} The ancient texts provide information about the world, its formation, the multiverses, the advancement of technology, the geography of Earth and Solar System and even informed us with the followers of Muhammed (Islam). https://www.supremeknowledge.org/islam/prophet-muhammad-in-bhavishya-purana/

=== Con ===

* ''Argument against'' The historical connection to the caste system. While considered mostly obsolete, caste identities can still influence social dynamics. 
* ''Argument against'' Ritualism may occur due to excessive emphasis placed on rituals, which may not always be understood or practiced meaningfully by practitioners, leading to a prioritisation of form over spiritual essence.   

== Christianity ==

=== Pro ===

* {{Argument for}} Christianity is a complete and well-established system of belief.

=== Con ===

* {{Argument against}} According to some, There is not enough evidence to support the most important assertions of Christianity, since Christianity is often at odds with empirical science&lt;ref&gt;{{Cite web|url=http://theconversation.com/even-setting-evolution-aside-basic-geology-disproves-creationism-40356|title=Even setting evolution aside, basic geology disproves creationism|last=Montgomery|first=David R.|website=The Conversation|language=en|access-date=2023-03-10}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.news24.com/news24/7-ways-god-is-debunked-by-the-sciences-20130826|title=7 ways God is debunked by the sciences|website=News24|language=en-US|access-date=2023-03-10}}&lt;/ref&gt;.
** {{Objection}} While some Biblical literalists (e.g. young-earth creationists) often deny science, many modern Christians are engaged in scientific study, like those on the website BioLogos.

== Buddhism ==

=== Pro ===

* {{Argument for}} Buddhism is the only religion which holds that the ease of human suffering (war, hunger etc.) is its principle purpose&lt;ref&gt;{{Cite web|url=https://www.pbs.org/edens/thailand/buddhism.htm|title=Basics of Buddhism|website=www.pbs.org|access-date=2023-03-10}}&lt;/ref&gt;.
* {{Argument for}} Buddhism focuses on expanding and improving one's own consciousness and being a good and healthy person all-round.
* {{Argument for}} Zen belief supports positive, ethical behaviors&lt;ref&gt;{{Cite web|url=https://zenstudies.org/about/ethical-guidelines/|title=Ethics Guidelines|website=Zen Studies|language=en-US|access-date=2023-03-10}}&lt;/ref&gt;.
** {{Objection}} Zen often encourages detachment from politics and social matters in the pursuit of personal enlightenment. Such apathy harms the world&lt;ref&gt;{{Cite web|url=http://hardcorezen.info/politics-a-buddhist-perspective/5706|title=Politics: A Buddhist Perspective {{!}} Hardcore Zen|language=en-US|access-date=2023-03-10}}&lt;/ref&gt;.

=== Con ===

* {{Argument against}} Buddhism promotes superstitious beliefs.
** {{Objection}} No example belief was stated. Thus, the above does not expose itself to refutation, making merely an existential statement, which are known to be either irrefutable/non-falsifiable to rather hard to verify. (The argument is so far as to be perhaps worthy of removal, or else serve as a basic argumentation teaching aid.)

== Islam ==

=== Pro ===

* {{Argument for}} Islamic epistemology marries the sources of knowledge to acquire truth, unlike other religions.&lt;ref&gt;{{Cite web|url=https://yaqeeninstitute.org/amp/justin-parrott/the-case-for-allahs-existence-in-the-quran-and-sunnah|title=The Case for Allah’s Existence in the Quran and Sunnah|website=Yaqeen Institute for Islamic Research|language=en|date=2017-02-27|author= Justin Parrott}}&lt;/ref&gt;
** {{Objection}} Which sources does it &quot;marry&quot;, specifically? What does it mean, specifically?
* {{Argument for}} Islamic epistemology is logically consistent &lt;ref&gt;{{Cite web|url=https://yaqeeninstitute.org/amp/justin-parrott/the-case-for-allahs-existence-in-the-quran-and-sunnah|title=The Case for Allah’s Existence in the Quran and Sunnah|website=Yaqeen Institute for Islamic Research|language=en|date=2017-02-27|author= Justin Parrott}}&lt;/ref&gt;.
** {{Objection}} The statement is not obvious and therefore needs some form of a proof. At a minimum, a link to a source arguing for consistency is to be provided.
** {{Objection}} Being logically consistent is no guarantee of goodness; rather, it would be a very basic prerequisite.
*** {{Objection}} Being logically consistent ''would'' be an argument for Islam, though not a complete and defining one, as certain other religions hold inconsistent beliefs. The concept of the [[w:Trinity|Christian trinity]] seems to defy logic. In the Trinity, although the Son, the Father, and the Holy Spirit are distinct persons despite being &quot;3 in 1&quot; - the roles that each character plays differ and contradict one another (for example, the Father is all-knowing but the Son is not)&lt;ref&gt;{{Cite book|url=https://www.bible.com/bible/compare/MRK.13.32-36|title=Mark 13:32-36|quote=But the exact day and hour? No one knows that, not even heaven’s angels, not even the Son. Only the Father. So keep a sharp lookout, for you don’t know the timetable. It’s like a man who takes a trip But of that day and that hour knoweth no man, no, not the angels which are in heaven, neither the Son, but the Father. Take ye heed, watch and pray: for ye know not when the time is. For the Son of ma But of that day or hour no one knows, not even the angels in heaven, nor the Son, but the Father alone. “Take heed, keep on the alert; for you do not know when the appointed time will come. It is like “No one knows when that day or time will be, not the angels in heaven, not even the Son. Only the Father knows. Be careful! Always be ready, because you don’t know when that time will be. It is like a But of that day or that hour knoweth no one, not even the angels in heaven, neither the Son, but the Father. Take ye heed, watch and pray: for ye know not when the time is. It is as when a man, sojour “But about that day or hour no one knows, not even the angels in heaven, nor the Son, but only the Father. Be on guard! Be alert! You do not know when that time will come. It’s like a man going away: “But of that day and hour no one knows, not even the angels in heaven, nor the Son, but only the Father. Take heed, watch and pray; for you do not know when the time is. It is like a man going to a fa But of that [exact] day or hour no one knows, not even the angels in heaven, nor the Son [in His humanity], but the Father alone. “Be on guard and stay constantly alert [ and pray ] ; for you do not k “However, no one knows the day or hour when these things will happen, not even the angels in heaven or the Son himself. Only the Father knows. And since you don’t know when that time will come, be on “Concerning that day and exact hour, no one knows when it will arrive, not the angels of heaven, not even the Son—only the Father knows. This is why you must be waiting, watching and praying, because “But concerning that day or that hour, no one knows, not even the angels in heaven, nor the Son, but only the Father. Be on guard, keep awake. For you do not know when the time will come. It is like a|language=en}}&lt;/ref&gt;. In Hinduism, religion can be interpreted based on the individual - therefore, certain Hindus can believe in monotheism while other Hindus believe in polytheism&lt;ref&gt;{{Cite journal|date=2023-10-16|title=God in Hinduism|url=https://en.wikipedia.org/w/index.php?title=God_in_Hinduism&amp;oldid=1180443952|journal=Wikipedia|language=en}}&lt;/ref&gt;.  In Islam, the concept of one, all-powerful God &amp; Muhammad being the last messenger has been well established in the Qur'an, the Sunnah, and the ''ijama'' [consensus] of the scholars. In fact, denying the creed of faith (shahada) takes one out of the fold of Islam &amp; is not considered a Muslim.
**** {{Objection}} The statement of the form &quot;X ''would be'' Y&quot; is consistent with &quot;X ''is not'' Y&quot;, and therefore, the above is no objection proper.
**** {{Objection}} Let us expound, then. Let us have a religion or an atheist philosophy that consistently aims at destruction of humankind (that, obviously, is not Islam). Such a religion or philosophy is consistent yet its aim is bad, and therefore, the religion or philosophy is bad. Therefore, consistency does not entail goodness.
***** {{Objection}} I agree with you that consistency does not entail goodness, but I'm saying that, compared to Christianity and Hinduism, is an advantage since we are able to reason &amp; ponder over a message that is consistent and a religion that does not contain contradictions in its theology or practice. For example, this could be used as an argument against Christianity in favor of Islam (&quot;worshipping 3 in 1 with contradictory features, making it seem paganistic&quot; vs. &quot;worshipping of one God and accepting Muhammad as his final messenger&quot;). Therefore, Islam's logical consistency is easy to accept and its lack of ''inconsistency'' cannot be used as an argument against the religion.
* {{Argument for}} Islam provides a logical and simple module to worship a higher entity: one All-Powerful, eternal God to which no entity can imitate.&lt;ref&gt;{{Cite journal|date=2023-10-30|title=Al-Ikhlas|url=https://en.wikipedia.org/w/index.php?title=Al-Ikhlas&amp;oldid=1182690024|journal=Wikipedia|language=en}}&lt;/ref&gt;

=== Con ===

* {{Argument against}} Islam's prophet spread the religion with violence, which is all too likely to inspire his followers to continue the violent practice.
** {{Objection}} There is some instruction toward violence in Old Testament as well. It is not clear that Islam is really worse than Judaism in this respect.
*** {{Objection}} The motion is not that Islam is better than Judaism but rather that it is best, among all religions. Therefore, it has to be better than Christianity, which does not depend solely on the Old Testament, but rather mitigates its arguable flaws in the New Testament, in the figure of Jesus Christ who, instead of using sword, allows himself to be killed in a crucifixion. And even if one decides to claim that the New Testament override of the old one is insufficient to stifle violence, one still has to show Islam to be better, as regards violence, than e.g. Buddhism and Hinduism.
** {{Objection}} Although violence itself ''was'' used to increase and maintain the power of the Arabs, this was not used to spread the religion of Islam &amp; the objection here implies that Muhammad used offensive violence to spread Islam. In fact, it is a popular claim that Islam 'spread by the sword', an untrue claim. The Muslims were oppressed by the polytheistic Arabs in Mecca after they were refusing to worship Arab idols (see [[w:Pre-Islamic Arabia]]), forcing them to migrate to Medina. After the peaceful [[w:Conquest_of_Mecca|conquest of Mecca in 630]] by the Muslim army, Islam spread abundantly throughout the Arabian peninsula. The Muslims were ''forced'' to fight the Byzantines after a Byzantine faction killed one of Muhammad's diplomats, leading to the [[w:Battle of Mu'tah]] &amp; the eventual conquest of Syria and the Levant. The Muslims fought the Sassanids, who were regarded as &quot;threatening&quot; to the Arabs&lt;ref&gt;Akbar Shah Najeebabadi, The history of Islam. B0006RTNB4.&lt;/ref&gt;. The Arabs invaded Egypt using violence (which was necessary to eliminate a threatening force), but the religion ''itself'' did not spread throughout the locals until over a 100 years later and was not spread through forced conversions or mass murder of non-Muslims&lt;ref&gt;{{Cite journal|date=2008-02-01|title=The great Arab conquests: how the spread of Islam changed the world we live in|url=http://dx.doi.org/10.5860/choice.45-3362|journal=Choice Reviews Online|volume=45|issue=06|pages=45–3362-45-3362|doi=10.5860/choice.45-3362|issn=0009-4978}}&lt;/ref&gt;. In fact, the Egyptians prefered Muslim rule over Byzantine [Christian] rule&lt;ref&gt;{{Cite web|url=https://www.britannica.com/place/Byzantine-Empire/The-successors-of-Justinian-565-610|title=Byzantine Empire - The successors of Justinian: 565–610 {{!}} Britannica|website=www.britannica.com|language=en|access-date=2023-11-22}}&lt;/ref&gt;. Many comments regarding oppression of the native Egyptian population were written by Catholic bishops, such as John of Nikiû, which were obviously marred with biases and inconsistencies. The religion of Islam spread amongst the Middle East &quot;voluntarily&quot; and was done mostly for &quot;economic advantage&quot;&lt;ref&gt;{{Cite book|title=A History of Islamic societies|last=Lapidus|first=Ira Marvin|date=1988|publisher=Cambridge university press|isbn=978-0-521-22552-6|location=Cambridge New York Melbourne}}&lt;/ref&gt;. Lastly, the Prophet advocated for tolerance of religion &amp; a multi-religious Islamic state. This is proven by the Qur'an&lt;ref&gt;{{Cite web|url=https://quran.com/al-kafirun|title=Surah Al-Kafirun - 1-6|website=Quran.com|language=en|access-date=2023-11-22}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://quran.com/al-baqarah/256|title=Surah Al-Baqarah - 256|website=Quran.com|language=en|access-date=2023-11-22}}&lt;/ref&gt; and the Sunnah&lt;ref&gt;{{Cite journal|last=El-Wakil|first=Ahmed|date=2019-09|title=“Whoever Harms a Dhimmī I Shall Be His Foe on the Day of Judgment”: An Investigation into an Authentic Prophetic Tradition and Its Origins from the Covenants|url=https://www.mdpi.com/2077-1444/10/9/516|journal=Religions|language=en|volume=10|issue=9|pages=516|doi=10.3390/rel10090516|issn=2077-1444}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=http://dx.doi.org/10.1163/1878-9781_ejiw_com_0012910|title=Khaybar|website=Encyclopedia of Jews in the Islamic World|access-date=2023-11-22}}&lt;/ref&gt;.

== Notes and references ==

{{Reflist}}

== See also ==

* [[Does God exist?]]
* [[Real Good Religion]] 

== External links ==

* [[Wikipedia:Major religious groups]]

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    <title>WikiJournal Preprints/CT Scan</title>
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      <text bytes="190008" xml:space="preserve">{{Article info
| first1 = Peerzada Mohammad
| last1 = Iflaq
| orcid1 = 0009-0005-4796-2375
| affiliation1 = Rayat Bahra University
| correspondence1 = peerzadaiflaq@gmail.com
| affiliations = institutes / affiliations
| et_al = https://en.wikipedia.org/w/index.php?title=CT_scan&amp;oldid=1156317872
| correspondence = email@address.com
| w1 = CT Scan
| from_w1 = true
| journal = WikiJournal of Medicine      &lt;!-- WikiJournal of Medicine, Science, or Humanities --&gt;
| license = &lt;!-- default is CC-BY --&gt;
| abstract = A computed tomography scan (usually abbreviated to CT scan; formerly called computed axial tomography scan or CAT scan) is a [[w:medical imaging|medical imaging]] technique used to obtain detailed internal images of the body. The personnel that perform CT scans are called [[w:radiographer|radiographer]]s or radiology technologists.

CT scanners use a rotating [[w:X-ray tube|X-ray tube]] and a row of detectors placed in a [[w:gantry (medical)|gantry]] to measure X-ray [[w:Attenuation#Radiography|attenuations]] by different tissues inside the body. The multiple [[w:X-ray|X-ray]] measurements taken from different angles are then processed on a computer using [[w:tomographic reconstruction|tomographic reconstruction]] algorithms to produce [[w:Tomography|tomographic]] (cross-sectional) images (virtual &quot;slices&quot;) of a body. CT scan can be used in patients with metallic implants or pacemakers, for whom [[w:magnetic resonance imaging|magnetic resonance imaging]] (MRI) is [[w:Contraindication|contraindicated]].

Compared to conventional [[:w:radiography|X-ray imaging]], CT scans provide detailed cross-sectional information of a specific area under examination, eliminates image superimposition which results in improved diagnostic capabilities. These tomographic images have proven to be valuable for accurate diagnosis and clinicopathological correlation in various [[w:medical condition|medical conditions]].
| keywords = &lt;!-- up to 6 keywords --&gt; CT Scan, Computed Tomography
}}

== Types ==

=== Generations of CT ===
The design and development of CT scanners went through multiple phases, which are collectively referred to as &quot;generations of CT.&quot;
[[File:First Generation CT Scan.svg|109x109px|1st Gen|frameless|right]]
The first-generation CT scanner, developed by Godfrey Hounsfield, also known as EMI scanner operated on the 'translate-rotate' principle. This system employed a pencil X-ray beam and two detectors, facilitating the acquisition of two views in the through-plane direction.&lt;ref&gt;{{Cite journal|last=Hounsfield|first=G. N.|date=1973-12|title=Computerized transverse axial scanning (tomography): Part 1. Description of system|url=http://www.birpublications.org/doi/abs/10.1259/0007-1285-46-552-1016|journal=The British Journal of Radiology|language=en|volume=46|issue=552|pages=1016–1022|doi=10.1259/0007-1285-46-552-1016|issn=0007-1285}}&lt;/ref&gt; The linear translation mechanism enabled the acquisition of 160 rays while the rotational movement captured 180 projections at 1° intervals, resulting in 28,800 rays for linear measurements.&lt;ref&gt;{{Cite journal|last=Ambrose|first=J.|last2=Hounsfield|first2=G.|date=1973-02|title=Computerized transverse axial tomography|url=https://pubmed.ncbi.nlm.nih.gov/4686818/|journal=The British Journal of Radiology|volume=46|issue=542|pages=148–149|issn=0007-1285|pmid=4686818}}&lt;/ref&gt; The CT scanner took about 4 minutes to acquire one slice and 15-20 minuntes to process the data.&lt;ref&gt;{{Cite journal|last=Friedland|first=G W|last2=Thurber|first2=B D|date=1996-12|title=The birth of CT.|url=https://www.ajronline.org/doi/10.2214/ajr.167.6.8956560|journal=American Journal of Roentgenology|language=en|volume=167|issue=6|pages=1365–1370|doi=10.2214/ajr.167.6.8956560|issn=0361-803X}}&lt;/ref&gt;&lt;ref name=&quot;auto7&quot;&gt;{{Cite journal|last=New|first=Paul F. J.|last2=Scott|first2=William R.|last3=Schnur|first3=James A.|last4=Davis|first4=Kenneth R.|last5=Taveras|first5=Juan M.|date=1974-01|title=Computerized Axial Tomography with the EMI Scanner|url=http://pubs.rsna.org/doi/10.1148/110.1.109|journal=Radiology|language=en|volume=110|issue=1|pages=109–123|doi=10.1148/110.1.109|issn=0033-8419}}&lt;/ref&gt; The first prototype model was installed in South London at Atkinson Morley's Hospital, and on October 1, 1971 first patient was scanned.&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Beckmann|first=E C|date=2006-01|title=CT scanning the early days|url=https://academic.oup.com/bjr/article/79/937/5-8/7443496|journal=The British Journal of Radiology|language=en|volume=79|issue=937|pages=5–8|doi=10.1259/bjr/29444122|issn=0007-1285}}&lt;/ref&gt; The first commercial scanner was installed at Mayo Clinic in 1973.&lt;ref name=&quot;auto7&quot;/&gt;
[[File:Second_Generation_CT_Scan.svg|left|frameless|111x111px]]
Following the initial development of the first-generation CT, EMI introduced &quot;CT 1010&quot;  an enhanced scanner in 1975 that eliminated the need for a waterbag. This system also employed &quot;translate-rotate&quot; configuration, but featured an upgraded setup with 8 detectors spanning 3 degrees. This enhancement allowed for a 3-degree rotation increment and required only 60 translations, significantly reducing the scan time to just 1 minute. This innovative design was subsequently referred to as the second-generation CT&lt;ref&gt;{{Cite journal|last=Schulz|first=Raymond A.|last2=Stein|first2=Jay A.|last3=Pelc|first3=Norbert J.|date=2021-10-29|title=How CT happened: the early development of medical computed tomography|url=https://www.spiedigitallibrary.org/journals/journal-of-medical-imaging/volume-8/issue-05/052110/How-CT-happened--the-early-development-of-medical-computed/10.1117/1.JMI.8.5.052110.full|journal=Journal of Medical Imaging|volume=8|issue=05|doi=10.1117/1.JMI.8.5.052110|issn=2329-4302|pmc=PMC8555965|pmid=34729383}}&lt;/ref&gt;. Subsequently, the detector count increased to 30, covering a range of 10 degrees and reducing the scan time to 20 seconds, as seen in the EMI 5000 series.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref&gt;{{Cite book|title=The essential physics of medical imaging|date=2012|publisher=Wolters Kluwer, Lippincott Williams &amp; Wilkins|isbn=978-0-7817-8057-5|editor-last=Bushberg|editor-first=Jerrold T.|edition=3. ed|location=Philadelphia}} p 370&lt;/ref&gt;
[[File:Third generation CT.svg|110x110px|Third generation CT|frameless|right]]
Third generation CT scanners used rotate-rotate configuration i.e. both the tube and the detectors rotated around the patient, employing a wide fan beam x-ray geometry and multiple detectors to collect the data.&lt;ref&gt;{{Cite journal|last=Chen|first=A. C.|last2=Berninger|first2=W. H.|last3=Redington|first3=R. W.|last4=Godbarsen|first4=R.|last5=Barrett|first5=D.|date=1976-12-23|editor-last=Cacak|editor-first=Robert K.|editor2-last=Carson|editor2-first=Paul L.|editor3-last=Dubuque|editor3-first=Gregory|editor4-last=Gray|editor4-first=Joel E.|editor5-last=Hendee|editor5-first=William R.|editor6-last=Rossi|editor6-first=Raymond P.|editor7-last=Haus|editor7-first=Arthur|title=Five-Second Fan Beam CT Scanner|url=http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1226195|pages=294–299|doi=10.1117/12.965426}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Goodenough|first=D. J.|last2=Weaver|first2=K. E.|date=1979-02|title=Overview of Computed Tomography|url=http://ieeexplore.ieee.org/document/4330458/|journal=IEEE Transactions on Nuclear Science|volume=26|issue=1|pages=1661–1667|doi=10.1109/TNS.1979.4330458|issn=0018-9499}}&lt;/ref&gt; In this generation of scanners, a singular detector element malfunction results in the erroneous recording of the corresponding ray in all projections, thereby inducing a ring artifact in the resultant images.&lt;ref&gt;{{Cite book|title=The essential physics of medical imaging|date=2012|publisher=Wolters Kluwer, Lippincott Williams &amp; Wilkins|isbn=978-0-7817-8057-5|editor-last=Bushberg|editor-first=Jerrold T.|edition=3. ed|location=Philadelphia}} p 371.&lt;/ref&gt; EMI 6000 belonged to the third generation of CT scanners.&lt;ref&gt;{{Cite journal|last=Mitchell|first=Se|last2=Clark|first2=Ra|date=1984-04-01|title=A comparison of computed tomography and sonography in choledocholithiasis|url=https://www.ajronline.org/doi/10.2214/ajr.142.4.729|journal=American Journal of Roentgenology|language=en|volume=142|issue=4|pages=729–733|doi=10.2214/ajr.142.4.729|issn=0361-803X}}&lt;/ref&gt;
[[File:Fourth Generation CT Scan.svg|left|frameless|112x112px|Fourth Generation CT Scan]]
The Fourth-generation CT Scanners, initially pioneered by A.S.&amp; E. Corporation,&lt;ref name=&quot;auto4&quot;&gt;{{Cite journal|last=Kak|first=A.C.|date=1979|title=Computerized tomography with X-ray, emission, and ultrasound sources|url=http://ieeexplore.ieee.org/document/1455709/|journal=Proceedings of the IEEE|volume=67|issue=9|pages=1245–1272|doi=10.1109/PROC.1979.11440|issn=0018-9219}}&lt;/ref&gt; employed detectors organized in a fixed ring comprising around 4800 individual detector elements. In this configuration, the X-ray tube generates a fan-beam X-ray and orbits around the patient. The ring artifact problem identified in Third-generation CT scanners was solved by this configuration. The EMI 7000 series similarly adhered to this principle.&lt;ref&gt;{{Cite book|title=The essential physics of medical imaging|date=2012|publisher=Wolters Kluwer, Lippincott Williams &amp; Wilkins|isbn=978-0-7817-8057-5|editor-last=Bushberg|editor-first=Jerrold T.|edition=3. ed|location=Philadelphia}} p 373.&lt;/ref&gt;

In Fifth generation CT, also know as electron beam computed tomography, both the x ray source and the detectors are stationary. This generation does not use conventional X-ray tube, rather employs a substantial tungsten arc (covering 210°) that surrounds the patient and directly faces the detector ring. An electron gun is utilized to guide and concentrate a rapid electron beam along the tungsten target ring within the gantry.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=SarDDwAAQBAJ&amp;q=ebct&amp;pg=PA6|title=Cardiovascular Computed Tomography|last=Stirrup|first=James|date=2020-01-02|publisher=Oxford University Press|isbn=978-0-19-880927-2|language=en}}&lt;/ref&gt; This type had a major advantage since sweep speeds can be much faster, allowing for less blurry imaging of moving structures, such as the heart and arteries.&lt;ref&gt;{{Cite journal|last1=Talisetti|first1=Anita|last2=Jelnin|first2=Vladimir|last3=Ruiz|first3=Carlos|last4=John|first4=Eunice|last5=Benedetti|first5=Enrico|last6=Testa|first6=Giuliano|last7=Holterman|first7=Ai-Xuan L.|last8=Holterman|first8=Mark J.|date=December 2004|title=Electron beam CT scan is a valuable and safe imaging tool for the pediatric surgical patient|journal=Journal of Pediatric Surgery|volume=39|issue=12|pages=1859–1862|doi=10.1016/j.jpedsurg.2004.08.024|issn=1531-5037|pmid=15616951}}&lt;/ref&gt; Fewer scanners of this design have been produced when compared with spinning tube types, mainly due to the higher cost associated with building a much larger X-ray tube and detector array and limited anatomical coverage.&lt;ref&gt;{{Cite journal|last=Retsky|first=Michael|date=31 July 2008|title=Electron beam computed tomography: Challenges and opportunities|journal=Physics Procedia|volume=1|issue=1|pages=149–154|bibcode=2008PhPro...1..149R|doi=10.1016/j.phpro.2008.07.090|doi-access=free}}&lt;/ref&gt;

=== Classification according to scanning method ===
Sequential CT, also known as step-and-shoot CT, is a scanning method in which the CT table moves stepwise. The process involves the table moving to a specific position, halting for the rotation and acquisition of a slice by the [[w:X-ray tube|X-ray tube]], followed by another incremental movement for the capture of subsequent slices. This method necessitates the table to pause during the slice acquisition, leading to increased scanning time due to interscan delays after each 360° rotation.&lt;ref&gt;{{Cite book |last=Terrier |first=F. |url=https://books.google.com/books?id=AV3wCAAAQBAJ&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA4&amp;dq=Sequential+CT+scan&amp;hl=en |title=Spiral CT of the Abdomen |last2=Grossholz |first2=M. |last3=Becker |first3=C. D. |date=2012-12-06 |publisher=Springer Science &amp; Business Media |isbn=978-3-642-56976-0 |page=4 |language=en}}&lt;/ref&gt; Before the introduction of slip ring technology, this scanning approach was commonly utilized. The need for the tube to return to its initial position after each rotation was essential to prevent cables connecting rotating components, such as the x-ray tube and detectors, from becoming entangled, resulting in prolonged interscan delays.[[File:Drawing of CT fan beam (left) and patient in a CT imaging system.gif|thumb|Drawing of CT fan beam and patient in a CT imaging system|223x223px]]
Spinning tube, commonly called [[w:Spiral computed tomography|spiral CT]], or helical CT, is an imaging technique in which an entire [[w:X-ray tube|X-ray tube]] is spun around the central axis of the area being scanned while the patient table is moving continuously.&lt;ref&gt;{{Cite journal|last=Fuchs|first=Theobald|last2=Kachelrieß|first2=Marc|last3=Kalender|first3=Willi A.|date=2000-11|title=Technical advances in multi–slice spiral CT|url=https://doi.org/10.1016/S0720-048X(00)00269-2|journal=European Journal of Radiology|volume=36|issue=2|pages=69–73|doi=10.1016/s0720-048x(00)00269-2|issn=0720-048X}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last1=Fishman |first1=Elliot K. |url=https://books.google.com/books?id=aWlrAAAAMAAJ&amp;q=spiral+ct |title=Spiral CT: Principles, Techniques, and Clinical Applications |last2=Jeffrey |first2=R. Brooke |date=1995 |publisher=Raven Press |isbn=978-0-7817-0218-8 |language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last=Hsieh |first=Jiang |url=https://books.google.com/books?id=JX__lLLXFHkC&amp;q=spiral+ct&amp;pg=PA265 |title=Computed Tomography: Principles, Design, Artifacts, and Recent Advances |date=2003 |publisher=SPIE Press |isbn=978-0-8194-4425-7 |page=265 |language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Crawford|first=Carl R.|last2=King|first2=Kevin F.|date=1990-11|title=Computed tomography scanning with simultaneous patient translation|url=https://aapm.onlinelibrary.wiley.com/doi/10.1118/1.596464|journal=Medical Physics|language=en|volume=17|issue=6|pages=967–982|doi=10.1118/1.596464|issn=0094-2405}}&lt;/ref&gt; Continuous scanning was made possible by slip ring technology. Slip rings provide an interface through a ring-and-brush arrangement, ensuring uninterrupted electrical connections. This eliminates the requirement for the X-ray tube to return to its initial position after each rotation, enabling continuous movement of the x-ray tube. The scanners with slip ring was introduced in 1987 by Siemens Medical Systems.&lt;ref&gt;{{Cite journal|last=Kalender|first=Willi A.|date=1996|editor-last=Vogl|editor-first=Thomas J.|editor2-last=Clauß|editor2-first=Wolfram|editor3-last=Li|editor3-first=Guo-Zhen|editor4-last=Yeon|editor4-first=Kyung Mo|title=Technical Foundations of Spiral Computed Tomography|url=https://link.springer.com/chapter/10.1007/978-3-642-79887-0_3|journal=Computed Tomography|language=en|location=Berlin, Heidelberg|publisher=Springer|pages=17–28|doi=10.1007/978-3-642-79887-0_3|isbn=978-3-642-79887-0}}&lt;/ref&gt;

=== Classifications according to the X-ray beam geometry ===
Pencil beam computed tomography employs a narrow, parallel X-ray beam geometry, while fan beam CT utilizes an X-ray beam that diverges outward from the radiation source.&lt;ref&gt;{{Cite journal|last=Horiba|first=Isao|last2=Yanaka|first2=Shigenobu|last3=Iwata|first3=Akira|last4=Suzumura|first4=Nobuo|date=1986-01|title=High‐resolution algorithm for fan beam‐CT system|url=https://onlinelibrary.wiley.com/doi/10.1002/scj.4690170703|journal=Systems and Computers in Japan|language=en|volume=17|issue=7|pages=19–29|doi=10.1002/scj.4690170703|issn=0882-1666}}&lt;/ref&gt;&lt;ref name=&quot;auto4&quot;/&gt;

Cone beam computed tomography (CBCT) uses a diverging cone shaped x ray beam for the generation of images. This type is particularly well-suited for dentistry and orthodontics due to its ability to achieve high-resolution imaging with voxel sizes as small as 0.1 mm. Moreover, it offers a considerable advantage over spiral CT by utilizing substantially lower levels of radiation.&lt;ref&gt;{{Cite journal|last=Larson|first=Brent E.|date=2012-04|title=Cone-beam computed tomography is the imaging technique of choice for comprehensive orthodontic assessment|url=https://doi.org/10.1016/j.ajodo.2012.02.009|journal=American Journal of Orthodontics and Dentofacial Orthopedics|volume=141|issue=4|pages=402–410|doi=10.1016/j.ajodo.2012.02.009|issn=0889-5406}}&lt;/ref&gt;

=== Classifications according to the detectors ===
A Single-row CT scanner utilizes a solitary row of detectors, enabling it to gather data for a single slice. Consequently, it is also referred to as Single Slice CT. Multi-row detector CT scanners are equipped with multiple rows of detectors. These detector rows acquire images simultaneously, enabling the rapid acquisition of multiple slices at once. MSCT scanners  offer enhanced image quality but come at the expense of increased radiation exposure compared to their single-slice CT. &lt;ref&gt;{{Cite journal|date=2008-08-20|title=Annals of the ICRP, Publication 102, Managing Patient Dose in Multi-Detector Computed Tomography (MDCT) Radiation Dose from Adult and Pediatric Multidetector Computed Tomography COMARE 12th Report, The Impact of Personally Initiated X-ray Computed Tomography Scanning for the Health Assessment of Asymptomatic Individuals The Psychology of Risk|url=http://dx.doi.org/10.1088/0952-4746/28/3/b01|journal=Journal of Radiological Protection|volume=28|issue=3|pages=435–441|doi=10.1088/0952-4746/28/3/b01|issn=0952-4746}}&lt;/ref&gt;

Photon-counting computed tomography  is a recent advancement in computed tomography that employs a photon-counting detector  to detect X-rays, registering the interactions of individual photons. Through monitoring the deposited energy in each interaction, the detectors capture an approximate energy spectrum.&lt;ref&gt;{{Cite journal|last=Willemink|first=Martin J.|last2=Persson|first2=Mats|last3=Pourmorteza|first3=Amir|last4=Pelc|first4=Norbert J.|last5=Fleischmann|first5=Dominik|date=2018-11|title=Photon-counting CT: Technical Principles and Clinical Prospects|url=http://pubs.rsna.org/doi/10.1148/radiol.2018172656|journal=Radiology|language=en|volume=289|issue=2|pages=293–312|doi=10.1148/radiol.2018172656|issn=0033-8419}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Leng|first=Shuai|last2=Bruesewitz|first2=Michael|last3=Tao|first3=Shengzhen|last4=Rajendran|first4=Kishore|last5=Halaweish|first5=Ahmed F.|last6=Campeau|first6=Norbert G.|last7=Fletcher|first7=Joel G.|last8=McCollough|first8=Cynthia H.|date=2019-05|title=Photon-counting Detector CT: System Design and Clinical Applications of an Emerging Technology|url=http://pubs.rsna.org/doi/10.1148/rg.2019180115|journal=RadioGraphics|language=en|volume=39|issue=3|pages=729–743|doi=10.1148/rg.2019180115|issn=0271-5333|pmc=PMC6542627|pmid=31059394}}&lt;/ref&gt;

Flat Panel CT represents a CT scanner which is characterized by the integration of flat panel detectors. These scanners present a significant advancement in volumetric coverage, facilitating comprehensive imaging of entire organs such as the heart, kidneys, or brain through a singular axial scan. &lt;ref&gt;{{Cite journal|last=Grasruck|first=M.|last2=Suess|first2=Ch|last3=Stierstorfer|first3=K.|last4=Popescu|first4=S.|last5=Flohr|first5=T.|date=2005-04-20|title=Evaluation of image quality and dose on a flat-panel CT-scanner|url=https://www.spiedigitallibrary.org/conference-proceedings-of-spie/5745/0000/Evaluation-of-image-quality-and-dose-on-a-flat-panel/10.1117/12.594583.full|journal=Medical Imaging 2005: Physics of Medical Imaging|publisher=SPIE|volume=5745|pages=179–188|doi=10.1117/12.594583}}&lt;/ref&gt;&lt;ref name=&quot;auto5&quot;&gt;{{Cite journal|last=Gupta|first=Rajiv|last2=Cheung|first2=Arnold C.|last3=Bartling|first3=Soenke H.|last4=Lisauskas|first4=Jennifer|last5=Grasruck|first5=Michael|last6=Leidecker|first6=Christianne|last7=Schmidt|first7=Bernhard|last8=Flohr|first8=Thomas|last9=Brady|first9=Thomas J.|date=2008-11|title=Flat-Panel Volume CT: Fundamental Principles, Technology, and Applications|url=http://pubs.rsna.org/doi/10.1148/rg.287085004|journal=RadioGraphics|language=en|volume=28|issue=7|pages=2009–2022|doi=10.1148/rg.287085004|issn=0271-5333}}&lt;/ref&gt;&lt;ref name=&quot;auto5&quot;/&gt;

=== Dual Energy CT ===
Dual Energy CT also known as Spectral CT is an advancement of Computed Tomography in which two energies are used to create two sets of data.&lt;ref&gt;{{Cite book |last=Johnson |first=Thorsten |url=https://books.google.com/books?id=Etvcnz0mjF4C&amp;newbks=0&amp;printsec=frontcover&amp;dq=dual+energy+ct&amp;hl=en |title=Dual Energy CT in Clinical Practice |last2=Fink |first2=Christian |last3=Schönberg |first3=Stefan O. |last4=Reiser |first4=Maximilian F. |date=2011-01-18 |publisher=Springer Science &amp; Business Media |isbn=978-3-642-01740-7 |language=en}}&lt;/ref&gt; A Dual Energy CT may employ Dual source, Single source with dual detector layer, Single source with energy switching methods to get two different sets of data.&lt;ref&gt;{{Cite book |last=Johnson |first=Thorsten |url=https://books.google.com/books?id=Etvcnz0mjF4C&amp;newbks=0&amp;printsec=frontcover&amp;dq=dual+energy+ct&amp;hl=en |title=Dual Energy CT in Clinical Practice |last2=Fink |first2=Christian |last3=Schönberg |first3=Stefan O. |last4=Reiser |first4=Maximilian F. |date=2011-01-18 |publisher=Springer Science &amp; Business Media |isbn=978-3-642-01740-7 |page=8 |language=en}}&lt;/ref&gt; It was commercially introduced in 2006.&lt;ref&gt;{{Cite journal|last=Schmidt|first=Bernhard|last2=Flohr|first2=Thomas|date=2020-11|title=Principles and applications of dual source CT|url=https://linkinghub.elsevier.com/retrieve/pii/S112017972030257X|journal=Physica Medica|language=en|volume=79|pages=36–46|doi=10.1016/j.ejmp.2020.10.014}}&lt;/ref&gt; Dual source CT is an advanced scanner with a two X-ray tube detector system, unlike conventional single tube systems.&lt;ref&gt;{{Cite book |last1=Carrascosa |first1=Patricia M. |url=https://books.google.com/books?id=wJ2oCgAAQBAJ&amp;q=dual+source+ct |title=Dual-Energy CT in Cardiovascular Imaging |last2=Cury |first2=Ricardo C. |last3=García |first3=Mario J. |last4=Leipsic |first4=Jonathon A. |date=2015-10-03 |publisher=Springer |isbn=978-3-319-21227-2 |language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal |last1=Schmidt |first1=Bernhard |last2=Flohr |first2=Thomas |date=2020-11-01 |title=Principles and applications of dual source CT |url=https://www.sciencedirect.com/science/article/pii/S112017972030257X |journal=Physica Medica |series=125 Years of X-Rays |language=en |volume=79 |pages=36–46 |doi=10.1016/j.ejmp.2020.10.014 |issn=1120-1797 |pmid=33115699 |s2cid=226056088}}&lt;/ref&gt; These two detector systems are mounted on a single gantry at 90° in the same plane.&lt;ref name=&quot;auto1&quot;&gt;{{Cite book |last1=Seidensticker |first1=Peter R. |url=https://books.google.com/books?id=oUtHea3ZnJ0C&amp;q=dual+source+ct |title=Dual Source CT Imaging |last2=Hofmann |first2=Lars K. |date=2008-05-24 |publisher=Springer Science &amp; Business Media |isbn=978-3-540-77602-4 |language=en}}&lt;/ref&gt;  This scanner allow fast scanning with higher temporal resolution by acquiring a full CT slice in only half a rotation. Fast imaging reduces motion blurring at high heart rates and potentially allowing for shorter breath-hold time. This is particularly useful for ill patients having difficulty holding their breath or unable to take heart-rate lowering medication.&lt;ref name=&quot;auto1&quot; /&gt;&lt;ref&gt;{{Cite journal |last1=Schmidt |first1=Bernhard |last2=Flohr |first2=Thomas |date=2020-11-01 |title=Principles and applications of dual source CT |url=https://www.physicamedica.com/article/S1120-1797(20)30257-X/abstract |journal=Physica Medica: European Journal of Medical Physics |language=English |volume=79 |pages=36–46 |doi=10.1016/j.ejmp.2020.10.014 |issn=1120-1797 |pmid=33115699 |s2cid=226056088}}&lt;/ref&gt;

Single Source with Energy switching is another mode of Dual energy CT in which a single tube is operated at two different energies by switching the energies frequently.&lt;ref&gt;{{Cite journal |last=Mahmood |first=Usman |last2=Horvat |first2=Natally |last3=Horvat |first3=Joao Vicente |last4=Ryan |first4=Davinia |last5=Gao |first5=Yiming |last6=Carollo |first6=Gabriella |last7=DeOcampo |first7=Rommel |last8=Do |first8=Richard K. |last9=Katz |first9=Seth |last10=Gerst |first10=Scott |last11=Schmidtlein |first11=C. Ross |last12=Dauer |first12=Lawrence |last13=Erdi |first13=Yusuf |last14=Mannelli |first14=Lorenzo |date=May 2018 |title=Rapid Switching kVp Dual Energy CT: Value of Reconstructed Dual Energy CT Images and Organ Dose Assessment in Multiphasic Liver CT Exams |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5918634/ |journal=European journal of radiology |volume=102 |pages=102–108 |doi=10.1016/j.ejrad.2018.02.022 |issn=0720-048X |pmc=5918634 |pmid=29685522}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal |last=Johnson |first=Thorsten R. C. |date=November 2012 |title=Dual-Energy CT: General Principles |url=https://www.ajronline.org/doi/10.2214/AJR.12.9116 |journal=American Journal of Roentgenology |language=en |volume=199 |issue=5_supplement |pages=S3–S8 |doi=10.2214/AJR.12.9116 |issn=0361-803X}}&lt;/ref&gt; Dual layer spectral CT is a sub-type in which the spectral data is obtained by using two separate scintillator layers. It consists of two detector layer in which one is on the top of another. The detector layer that is closer to the x ray tube detects the low energy x rays and lets the high energy x rays to pass to the layer that is below. The high energy x rays are detected by the second layer.&lt;ref&gt;{{Cite journal|last=Ehn|first=Sebastian|last2=Sellerer|first2=Thorsten|last3=Muenzel|first3=Daniela|last4=Fingerle|first4=Alexander A.|last5=Kopp|first5=Felix|last6=Duda|first6=Manuela|last7=Mei|first7=Kai|last8=Renger|first8=Bernhard|last9=Herzen|first9=Julia|date=2018-01|title=Assessment of quantification accuracy and image quality of a full‐body dual‐layer spectral CT system|url=https://onlinelibrary.wiley.com/doi/10.1002/acm2.12243|journal=Journal of Applied Clinical Medical Physics|language=en|volume=19|issue=1|pages=204–217|doi=10.1002/acm2.12243|issn=1526-9914|pmc=PMC5768037|pmid=29266724}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Rassouli|first=Negin|last2=Etesami|first2=Maryam|last3=Dhanantwari|first3=Amar|last4=Rajiah|first4=Prabhakar|date=2017-12-01|title=Detector-based spectral CT with a novel dual-layer technology: principles and applications|url=https://doi.org/10.1007/s13244-017-0571-4|journal=Insights into Imaging|language=en|volume=8|issue=6|pages=589–598|doi=10.1007/s13244-017-0571-4|issn=1869-4101|pmc=PMC5707218|pmid=28986761}}&lt;/ref&gt;

=== Hybrid CT imaging ===
Hybrid imaging involves integrating two or more imaging modalities to create a novel technique. This fusion leverages the inherent strengths of the combined imaging technologies, resulting in the emergence of a more potent and advanced modality.[[File:Petct1.jpg|thumb|PET-CT scan of chest|161x161px]]
Positron emission tomography–computed tomography is a hybrid CT modality which combines, in a single gantry, a [[w:positron emission tomography|positron emission tomography]] (PET) scanner and an x-ray computed tomography scanner, to acquire sequential images from both devices in the same session, which are combined into a single superposed ([[w:Image registration|co-registered]]) image. Thus, [[w:functional imaging|functional imaging]] obtained by PET, which depicts the spatial distribution of metabolic or biochemical activity in the body can be more precisely aligned or correlated with anatomic imaging obtained by CT scanning.&lt;ref&gt;{{Cite journal |last=Blodgett |first=Todd M. |last2=Meltzer |first2=Carolyn C. |last3=Townsend |first3=David W. |date=February 2007 |title=PET/CT: form and function |url=https://pubmed.ncbi.nlm.nih.gov/17255408/#:~:text=CT%20is%20complementary%20in%20the,identify%20and%20localize%20functional%20abnormalities. |journal=Radiology |volume=242 |issue=2 |pages=360–385 |doi=10.1148/radiol.2422051113 |issn=0033-8419 |pmid=17255408}}&lt;/ref&gt; PET-CT gives both anatomical and functional details of an organ under examination and is helpful in detecting different type of cancers.&lt;ref&gt;{{Cite journal |last=Ciernik |first=I.Frank |last2=Dizendorf |first2=Elena |last3=Baumert |first3=Brigitta G |last4=Reiner |first4=Beatrice |last5=Burger |first5=Cyrill |last6=Davis |first6=J.Bernard |last7=Lütolf |first7=Urs M |last8=Steinert |first8=Hans C |last9=Von Schulthess |first9=Gustav K |date=November 2003 |title=Radiation treatment planning with an integrated positron emission and computer tomography (PET/CT): a feasibility study |url=https://doi.org/10.1016/S0360-3016(03)00346-8 |journal=International Journal of Radiation Oncology*Biology*Physics |volume=57 |issue=3 |pages=853–863 |doi=10.1016/s0360-3016(03)00346-8 |issn=0360-3016}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal |last=Ul-Hassan |first=Fahim |last2=Cook |first2=Gary J |date=August 2012 |title=PET/CT in oncology |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4952129/ |journal=Clinical Medicine |volume=12 |issue=4 |pages=368–372 |doi=10.7861/clinmedicine.12-4-368 |issn=1470-2118 |pmc=4952129 |pmid=22930885}}&lt;/ref&gt; Hybrid PET-CT systems have become more effective with the integration of anatomical details from CT scans. This integration allows for the creation of an attenuation correction map, which helps refine PET images. These advancements have notably reduced examination duration, increased diagnostic accuracy, and instilled greater confidence in the accuracy of diagnoses.&lt;ref&gt;{{Cite journal|last=Veit-Haibach|first=Patrick|last2=Luczak|first2=Christopher|last3=Wanke|first3=Isabel|last4=Fischer|first4=Markus|last5=Egelhof|first5=Thomas|last6=Beyer|first6=Thomas|last7=Dahmen|first7=Gerlinde|last8=Bockisch|first8=Andreas|last9=Rosenbaum|first9=Sandra|date=2007-12-01|title=TNM staging with FDG-PET/CT in patients with primary head and neck cancer|url=https://doi.org/10.1007/s00259-007-0564-5|journal=European Journal of Nuclear Medicine and Molecular Imaging|language=en|volume=34|issue=12|pages=1953–1962|doi=10.1007/s00259-007-0564-5|issn=1619-7089}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Sonni|first=Ida|last2=Baratto|first2=Lucia|last3=Park|first3=Sonya|last4=Hatami|first4=Negin|last5=Srinivas|first5=Shyam|last6=Davidzon|first6=Guido|last7=Gambhir|first7=Sanjiv Sam|last8=Iagaru|first8=Andrei|date=2018-04-18|title=Initial experience with a SiPM-based PET/CT scanner: influence of acquisition time on image quality|url=https://doi.org/10.1186/s40658-018-0207-x|journal=EJNMMI Physics|volume=5|issue=1|pages=9|doi=10.1186/s40658-018-0207-x|issn=2197-7364|pmc=PMC5904089|pmid=29666972}}&lt;/ref&gt; In oncology, studies show that using PET-CT together is better for accurately staging and restaging than using CT or PET alone.&lt;ref&gt;{{Cite journal|last=Ben-Haim|first=Simona|last2=Ell|first2=Peter|date=2009-01-01|title=18F-FDG PET and PET/CT in the Evaluation of Cancer Treatment Response|url=https://jnm.snmjournals.org/content/50/1/88|journal=Journal of Nuclear Medicine|language=en|volume=50|issue=1|pages=88–99|doi=10.2967/jnumed.108.054205|issn=0161-5505|pmid=19139187}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Czernin|first=Johannes|last2=Allen-Auerbach|first2=Martin|last3=Schelbert|first3=Heinrich R.|date=2007-01-01|title=Improvements in Cancer Staging with PET/CT: Literature-Based Evidence as of September 2006|url=https://jnm.snmjournals.org/content/48/1_suppl/78S|journal=Journal of Nuclear Medicine|language=en|volume=48|issue=1 suppl|pages=78S–88S|issn=0161-5505|pmid=17204723}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Ul-Hassan|first=Fahim|last2=Cook|first2=Gary J|date=2012-08|title=PET/CT in oncology|url=https://www.rcpjournals.org/lookup/doi/10.7861/clinmedicine.12-4-368|journal=Clinical Medicine|language=en|volume=12|issue=4|pages=368–372|doi=10.7861/clinmedicine.12-4-368|issn=1470-2118|pmc=PMC4952129|pmid=22930885}}&lt;/ref&gt;

Single photon emission computed tomography- computed tomography also known as SPECT-CT is a hybrid imaging modality, used in Nuclear Medicine which combines a [[w:Single-photon emission computed tomography|SPECT]] scanner and a CT scanner into one machine. This hybrid modality was first introduced commercially in 1999&lt;ref name=&quot;auto6&quot;&gt;{{Cite journal|last=Ritt|first=P.|last2=Sanders|first2=J.|last3=Kuwert|first3=T.|date=2014-12-01|title=SPECT/CT technology|url=https://doi.org/10.1007/s40336-014-0086-7|journal=Clinical and Translational Imaging|language=en|volume=2|issue=6|pages=445–457|doi=10.1007/s40336-014-0086-7|issn=2281-7565}}&lt;/ref&gt; SPECT-CT uses a radiotracer for evaluation of function details and x rays anatomical details. These image sets are then coregistered to allow for a comprehensive understanding of the relationship between physiological function and anatomical structures, aiding in more accurate and reliable diagnostic evaluations.&lt;ref name=&quot;auto6&quot;/&gt;

Angio-CT is a hybrid machine which combines the fluoroscopic angiographic imaging and cross-sectional imaging of CT. These systems offer an integrated approach that combines the benefits of conventional angiography with the imaging capabilities of CT technology.&lt;ref name=&quot;auto2&quot;&gt;{{Cite journal|last=Taiji|first=Ryosuke|last2=Lin|first2=Ethan Y.|last3=Lin|first3=Yuan-Mao|last4=Yevich|first4=Steven|last5=Avritscher|first5=Rony|last6=Sheth|first6=Rahul A.|last7=Ruiz|first7=Joseph R.|last8=Jones|first8=A. Kyle|last9=Chintalapani|first9=Gouthami|date=2021-09-01|title=Combined Angio-CT Systems: A Roadmap Tool for Precision Therapy in Interventional Oncology|url=http://pubs.rsna.org/doi/10.1148/rycan.2021210039|journal=Radiology: Imaging Cancer|language=en|volume=3|issue=5|pages=e210039|doi=10.1148/rycan.2021210039|issn=2638-616X|pmc=PMC8489448|pmid=34559007}}&lt;/ref&gt; This hybrid imaging modality was introduced by Yasuaki Arai in 1992.&lt;ref&gt;{{Cite journal|last=Yoshitaka|first=Inaba|last2=Yasuaki|first2=Arai|last3=Yoshito|first3=Takeuchi|last4=Hideyuki|first4=Takeda|last5=Toyohiro|first5=Ota|last6=Satoru|first6=Sueyoshi|last7=Takuji|first7=Yamagami|last8=Kazuhiko|first8=Ohashi|last9=K|first9=Yun|date=1996|title=New Diagnostic Imagings for IVR. Clinical Effectiveness of a Newly Developed Interventional-CT system.|url=https://jglobal.jst.go.jp/en/detail?JGLOBAL_ID=200902150551428412|journal=IVR|language=en|volume=11|issue=1|pages=43–49|issn=1340-4520}}&lt;/ref&gt;&lt;ref name=&quot;auto2&quot;/&gt;

== Medical use ==
Since its introduction in the 1970s,&lt;ref&gt;{{Cite book |last1=Curry |first1=Thomas S. |url=https://books.google.com/books?id=W2PrMwHqXl0C |title=Christensen's Physics of Diagnostic Radiology |last2=Dowdey |first2=James E. |last3=Murry |first3=Robert C. |date=1990 |publisher=Lippincott Williams &amp; Wilkins |isbn=978-0-8121-1310-5 |pages=289 |language=en}}&lt;/ref&gt; CT has become an important tool in [[w:medical imaging|medical imaging]] to supplement conventional [[w:X-ray|X-ray]] imaging and [[w:medical ultrasonography|medical ultrasonography]]. It has more recently been used for [[w:preventive medicine|preventive medicine]] or [[w:screening (medicine)|screening]] for disease, for example, [[w:Virtual colonoscopy|CT colonography]] for people with a high risk of [[w:colon cancer|colon cancer]], or full-motion heart scans for people with a high risk of heart disease. The use of CT scans has increased dramatically over the last two decades in many countries.&lt;ref name=&quot;Smith2009&quot;&gt;{{Cite journal |vauthors=Smith-Bindman R, Lipson J, Marcus R, Kim KP, Mahesh M, Gould R, Berrington de González A, [[Diana Miglioretti|Miglioretti DL]] |date=December 2009 |title=Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer |journal=Arch. Intern. Med. |volume=169 |issue=22 |pages=2078–86 |doi=10.1001/archinternmed.2009.427 |pmc=4635397 |pmid=20008690}}&lt;/ref&gt;  An estimated 72 million scans were performed in the United States in 2007 and more than 80 million in 2015.&lt;ref name=&quot;Berrington2009&quot;&gt;{{Cite journal |vauthors=Berrington de González A, Mahesh M, Kim KP, Bhargavan M, Lewis R, Mettler F, Land C |date=December 2009 |title=Projected cancer risks from computed tomographic scans performed in the United States in 2007 |journal=Arch. Intern. Med. |volume=169 |issue=22 |pages=2071–7 |doi=10.1001/archinternmed.2009.440 |pmc=6276814 |pmid=20008689}}&lt;/ref&gt;&lt;ref&gt;{{Cite web |title=Dangers of CT Scans and X-Rays – Consumer Reports |url=https://www.consumerreports.org/cro/magazine/2015/01/the-surprising-dangers-of-ct-sans-and-x-rays/index.htm |access-date=16 May 2018}}&lt;/ref&gt;

=== Diagnostic ===

==== Head &amp; Neck Imaging ====
[[File:CT of a normal brain (thumbnail).png|thumb|218x218px|CT Head of a normal brain: Sagittal (top), Coronal (bottom left), Axial (bottom right)]]
CT scan remains the cornerstone imaging modality for the initial evaluation and subsequent management of patients with acute traumatic brain injury due to its rapid acquisition time and high sensitivity for detecting hemorrhagic complications, such as intraparenchymal hematomas and subdural hemorrhages.&lt;ref&gt;{{Cite journal|last=Schweitzer|first=Andrew D.|last2=Niogi|first2=Sumit N.|last3=Whitlow|first3=Christopher J.|last4=Tsiouris|first4=A. John|date=2019-10|title=Traumatic Brain Injury: Imaging Patterns and Complications|url=http://pubs.rsna.org/doi/10.1148/rg.2019190076|journal=RadioGraphics|language=en|volume=39|issue=6|pages=1571–1595|doi=10.1148/rg.2019190076|issn=0271-5333}}&lt;/ref&gt; CT scan of the head is typically used to detect [[w:infarction|infarction]] ([[w:stroke|stroke]]), [[w:Neoplasm|tumors]], [[w:calcification|calcification]]s, [[w:haemorrhage|haemorrhage]].&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=pUVwDwAAQBAJ&amp;q=CT+scanning+of+the+head+is+typically+used+to+detect&amp;pg=PA389|title=Critical Care Transport|last1=Surgeons (AAOS)|first1=American Academy of Orthopaedic|last2=Physicians (ACEP)|first2=American College of Emergency|last3=UMBC|date=2017-03-20|publisher=Jones &amp; Bartlett Learning|isbn=978-1-284-04099-9|page=389|language=en}}&lt;/ref&gt; Tumors can be detected by the swelling and anatomical distortion they cause, or by surrounding edema. CT scanning of the head is also used in CT-[[w:image guided surgery|guided]] [[w:stereotactic surgery|stereotactic surgery]] and [[w:radiosurgery|radiosurgery]] for treatment of intracranial tumors, [[w:arteriovenous malformation|arteriovenous malformation]]s, and other surgically treatable conditions using a device known as the [[w:N-localizer|N-localizer]].&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=ioxongEACAAJ|title=Image-Guided Neurosurgery|last=Galloway|first=RL Jr.|publisher=Elsevier|year=2015|isbn=978-0-12-800870-6|editor-last=Golby|editor-first=AJ|location=Amsterdam|pages=3–4|chapter=Introduction and Historical Perspectives on Image-Guided Surgery}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=uEghr21XY6wC|title=Principles and Practice of Stereotactic Radiosurgery|last1=Tse|first1=VCK|last2=Kalani|first2=MYS|last3=Adler|first3=JR|publisher=Springer|year=2015|isbn=978-0-387-71070-9|editor-last=Chin|editor-first=LS|location=New York|page=28|chapter=Techniques of Stereotactic Localization|editor-last2=Regine|editor-first2=WF}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|title=Stereotactic Radiosurgery and Stereotactic Body Radiation Therapy|last1=Saleh|first1=H|last2=Kassas|first2=B|publisher=CRC Press|year=2015|isbn=978-1-4398-4198-3|editor-last=Benedict|editor-first=SH|location=Boca Raton|pages=156–159|chapter=Developing Stereotactic Frames for Cranial Treatment|editor-last2=Schlesinger|editor-first2=DJ|editor-last3=Goetsch|editor-first3=SJ|editor-last4=Kavanagh|editor-first4=BD|chapter-url=https://books.google.com/books?id=Pm3RBQAAQBAJ&amp;q=Developing+Stereotactic+Frames+for+Cranial+Treatment&amp;pg=PA153}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=mAN3MAEACAAJ&amp;q=0444534970|title=Brain Stimulation|last1=Khan|first1=FR|last2=Henderson|first2=JM|journal=Handbook of Clinical Neurology|publisher=Elsevier|year=2013|isbn=978-0-444-53497-2|editor-last=Lozano|editor-first=AM|volume=116|location=Amsterdam|pages=28–30|chapter=Deep Brain Stimulation Surgical Techniques|doi=10.1016/B978-0-444-53497-2.00003-6|pmid=24112882|editor-last2=Hallet|editor-first2=M}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=cnF-2KCeR1sC&amp;q=Textbook+of+Stereotactic+and+Functional+Neurosurgery|title=Textbook of Stereotactic and Functional Neurosurgery|last=Arle|first=J|publisher=Springer-Verlag|year=2009|isbn=978-3-540-69959-0|editor-last=Lozano|editor-first=AM|location=Berlin|pages=456–461|chapter=Development of a Classic: the Todd-Wells Apparatus, the BRW, and the CRW Stereotactic Frames|editor-last2=Gildenberg|editor-first2=PL|editor-last3=Tasker|editor-first3=RR}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=June 2012|title=Invention of the N-localizer for stereotactic neurosurgery and its use in the Brown-Roberts-Wells stereotactic frame|journal=Neurosurgery|volume=70|issue=2 Supplement Operative|pages=173–176|doi=10.1227/NEU.0b013e318246a4f7|pmid=22186842|vauthors=Brown RA, Nelson JA|s2cid=36350612}}&lt;/ref&gt;

[[w:Contrast CT|Contrast CT]] is generally the initial study of choice for [[w:neck mass|neck mass]]es in adults.&lt;ref&gt;{{Cite journal|last=Das|first=Riya|last2=Sarkar|first2=Tanmoy|last3=Verma|first3=Sweta|date=2022-12|title=A Case Series on Unusual Neck Masses|url=https://link.springer.com/10.1007/s12070-021-02866-5|journal=Indian Journal of Otolaryngology and Head &amp; Neck Surgery|language=en|volume=74|issue=S3|pages=5531–5535|doi=10.1007/s12070-021-02866-5|issn=2231-3796|pmc=PMC8461437|pmid=34584897}}&lt;/ref&gt; [[w:Computed tomography of the thyroid|CT of the thyroid]] plays an important role in the evaluation of [[w:thyroid cancer|thyroid cancer]].&lt;ref name=&quot;Saeedan2016&quot;&gt;{{Cite journal|last1=Bin Saeedan|first1=Mnahi|last2=Aljohani|first2=Ibtisam Musallam|last3=Khushaim|first3=Ayman Omar|last4=Bukhari|first4=Salwa Qasim|last5=Elnaas|first5=Salahudin Tayeb|year=2016|title=Thyroid computed tomography imaging: pictorial review of variable pathologies|journal=Insights into Imaging|volume=7|issue=4|pages=601–617|doi=10.1007/s13244-016-0506-5|issn=1869-4101|pmc=4956631|pmid=27271508}}&lt;/ref&gt; CT scan often incidentally finds thyroid abnormalities, and so is often the preferred investigation modality for thyroid abnormalities.&lt;ref name=&quot;Saeedan2016&quot; /&gt;

==== Body Imaging ====
A CT scan can be used for detecting both acute and chronic changes in the [[w:Parenchyma#Lung parenchyma|lung parenchyma]], the tissue of the [[w:lung|lung]]s.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=rQlDDwAAQBAJ|title=Computed Tomography of the Lung|publisher=Springer Berlin Heidelberg|year=2007|isbn=978-3-642-39518-5|pages=40, 47}}&lt;/ref&gt; It is particularly relevant here because normal two-dimensional X-rays do not show such defects. A variety of techniques are used, depending on the suspected abnormality. For evaluation of chronic interstitial processes such as [[w:Pneumatosis#Lungs|emphysema]], and [[w:Pulmonary fibrosis#|fibrosis]],&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=VKATAQAAMAAJ&amp;q=ct+of+lungs|title=High-resolution CT of the Lung|publisher=Lippincott Williams &amp; Wilkins|year=2009|isbn=978-0-7817-6909-9|pages=81,568}}&lt;/ref&gt; thin sections with high spatial frequency reconstructions are used; often scans are performed both on inspiration and expiration. This special technique is called [[w:high resolution CT|high resolution CT]] that produces a sampling of the lung, and not continuous images.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=QjouDwAAQBAJ&amp;q=HRCT|title=Specialty Imaging: HRCT of the Lung E-Book|last1=Martínez-Jiménez|first1=Santiago|last2=Rosado-de-Christenson|first2=Melissa L.|last3=Carter|first3=Brett W.|date=2017-07-22|publisher=Elsevier Health Sciences|isbn=978-0-323-52495-7|language=en}}&lt;/ref&gt;

CT is an accurate technique for diagnosis of [[w:Human abdomen|abdominal]] diseases like [[w:Crohn's disease|Crohn's disease]],&lt;ref&gt;{{Cite journal|last1=Furukawa|first1=Akira|last2=Saotome|first2=Takao|last3=Yamasaki|first3=Michio|last4=Maeda|first4=Kiyosumi|last5=Nitta|first5=Norihisa|last6=Takahashi|first6=Masashi|last7=Tsujikawa|first7=Tomoyuki|last8=Fujiyama|first8=Yoshihide|last9=Murata|first9=Kiyoshi|date=2004-05-01|title=Cross-sectional Imaging in Crohn Disease|journal=RadioGraphics|volume=24|issue=3|pages=689–702|doi=10.1148/rg.243035120|issn=0271-5333|pmid=15143222|doi-access=free|last10=Sakamoto|first10=Tsutomu}}&lt;/ref&gt; GIT bleeding, and diagnosis and staging of cancer, as well as follow-up after cancer treatment to assess response.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=r3uK7sSZUmcC|title=CT of the Acute Abdomen|publisher=Springer Berlin Heidelberg|year=2011|isbn=978-3-540-89232-8|pages=37}}&lt;/ref&gt; It is commonly used to investigate [[w:acute abdominal pain|acute abdominal pain]].&lt;ref&gt;{{Cite book|title=Diseases of the Abdomen and Pelvis|last1=Jay P Heiken|last2=Douglas S Katz|publisher=Springer Milan|year=2014|isbn=9788847056596|editor-last=J. Hodler|page=3|chapter=Emergency Radiology of the Abdomen and Pelvis: Imaging of the Nontraumatic and Traumatic Acute Abdomen|editor-last2=R. A. Kubik-Huch|editor-last3=G. K. von Schulthess|editor-last4=Ch. L. Zollikofer|chapter-url=https://books.google.com/books?id=CSy5BQAAQBAJ&amp;pg=PA3}}&lt;/ref&gt; Non-enhanced computed tomography is today the gold standard for diagnosing [[w:Kidney stone disease|urinary stones]].&lt;ref&gt;{{Cite book|url=https://uroweb.org/guidelines/urolithiasis|title=EAU Guidelines on Urolithiasis|last1=Skolarikos|first1=A|last2=Neisius|first2=A|last3=Petřík|first3=A|last4=Somani|first4=B|last5=Thomas|first5=K|last6=Gambaro|first6=G|date=March 2022|publisher=[[European Association of Urology]]|isbn=978-94-92671-16-5|location=Amsterdam}}&lt;/ref&gt; The size, volume and density of stones can be estimated to help clinicians guide further treatment; size is especially important in predicting spontaneous passage of a stone.&lt;ref&gt;{{Cite journal|last1=Miller|first1=Oren F.|last2=Kane|first2=Christopher J.|date=September 1999|title=Time to stone passage for observed ureteral calculi: a guide for patient education|journal=Journal of Urology|volume=162|issue=3 Part 1|pages=688–691|doi=10.1097/00005392-199909010-00014|pmid=10458343}}&lt;/ref&gt;

==== Musculoskeletal Imaging ====
CT scan is widely used for imaging of muscluloskeltal. For the [[w:axial skeleton|axial skeleton]] and [[w:Limb (anatomy)|extremities]], CT is often used to image complex [[w:fracture (bone)|fractures]], especially ones around joints, because of its ability to reconstruct the area of interest in multiple planes.&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/174_2017_25|title=Clinical Application of Musculoskeletal CT: Trauma, Oncology, and Postsurgery|last=Gondim Teixeira|first=Pedro Augusto|last2=Blum|first2=Alain|date=2019|publisher=Springer International Publishing|isbn=978-3-319-42586-3|editor-last=Nikolaou|editor-first=Konstantin|series=Medical Radiology|location=Cham|pages=1079–1105|language=en|doi=10.1007/174_2017_25|editor-last2=Bamberg|editor-first2=Fabian|editor-last3=Laghi|editor-first3=Andrea|editor-last4=Rubin|editor-first4=Geoffrey D.}}&lt;/ref&gt; Fractures, ligamentous injuries, and [[w:Dislocation (medicine)|dislocations]] can easily be recognized with a 0.2&amp;nbsp;mm resolution.&lt;ref&gt;{{Cite web|url=http://orthoinfo.aaos.org/topic.cfm?topic=A00391|title=Ankle Fractures|website=orthoinfo.aaos.org|publisher=American Association of Orthopedic Surgeons|archive-url=https://web.archive.org/web/20100530103553/http://orthoinfo.aaos.org/topic.cfm?topic=A00391|archive-date=30 May 2010|access-date=30 May 2010|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Buckwalter, Kenneth A.|display-authors=etal|date=11 September 2000|title=Musculoskeletal Imaging with Multislice CT|journal=American Journal of Roentgenology|volume=176|issue=4|pages=979–986|doi=10.2214/ajr.176.4.1760979|pmid=11264094}}&lt;/ref&gt; With modern dual-energy CT scanners, new areas of use have been established, such as aiding in the diagnosis of [[w:gout|gout]].&lt;ref&gt;{{Cite journal|last1=Ramon|first1=André|last2=Bohm-Sigrand|first2=Amélie|last3=Pottecher|first3=Pierre|last4=Richette|first4=Pascal|last5=Maillefert|first5=Jean-Francis|last6=Devilliers|first6=Herve|last7=Ornetti|first7=Paul|date=2018-03-01|title=Role of dual-energy CT in the diagnosis and follow-up of gout: systematic analysis of the literature|journal=Clinical Rheumatology|volume=37|issue=3|pages=587–595|doi=10.1007/s10067-017-3976-z|issn=0770-3198|pmid=29350330|s2cid=3686099}}&lt;/ref&gt;

==== Perfusion Imaging ====
[[File:CT perfusion in M1 artery occlusion.png|thumb|CT Perfusion images of Brain with Time to Peak, Cerebral blood volume.|195x195px]]CT perfusion imaging is a specific form of CT to assess flow through [[w:blood vessel|blood vessel]]s whilst injecting a [[w:contrast agent|contrast agent]].&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last1=Wittsack|first1=H.-J.|last2=Wohlschläger|first2=A.M.|last3=Ritzl|first3=E.K.|last4=Kleiser|first4=R.|last5=Cohnen|first5=M.|last6=Seitz|first6=R.J.|last7=Mödder|first7=U.|date=2008-01-01|title=CT-perfusion imaging of the human brain: Advanced deconvolution analysis using circulant singular value decomposition|journal=Computerized Medical Imaging and Graphics|language=en|volume=32|issue=1|pages=67–77|doi=10.1016/j.compmedimag.2007.09.004|issn=0895-6111|pmid=18029143}}&lt;/ref&gt; Blood flow, blood transit time, and organ blood volume, can all be calculated with reasonable [[w:sensitivity and specificity|sensitivity and specificity]].&lt;ref name=&quot;:0&quot; /&gt; This type of CT may be used on the [[w:heart|heart]], although sensitivity and specificity for detecting abnormalities are still lower than for other forms of CT.&lt;ref&gt;{{Cite journal|last1=Williams|first1=M.C.|last2=Newby|first2=D.E.|date=2016-08-01|title=CT myocardial perfusion imaging: current status and future directions|journal=Clinical Radiology|language=en|volume=71|issue=8|pages=739–749|doi=10.1016/j.crad.2016.03.006|issn=0009-9260|pmid=27091433}}&lt;/ref&gt; This may also be used on the [[w:brain|brain]], where CT perfusion imaging can often detect poor brain perfusion well before it is detected using a conventional spiral CT scan.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last1=Donahue|first1=Joseph|last2=Wintermark|first2=Max|date=2015-02-01|title=Perfusion CT and acute stroke imaging: Foundations, applications, and literature review|journal=Journal of Neuroradiology|language=en|volume=42|issue=1|pages=21–29|doi=10.1016/j.neurad.2014.11.003|issn=0150-9861|pmid=25636991}}&lt;/ref&gt; This is better for [[w:stroke|stroke]] diagnosis than other CT types.&lt;ref name=&quot;:1&quot; /&gt;

==== Cardiac Imaging ====
[[File:SADDLE PE.JPG|thumb|CTPA, demonstrating a saddle [[w:pulmonary embolism|embolus]] (dark horizontal line) occluding the [[w:pulmonary artery|pulmonary arteries]] (bright white triangle)|191x191px]]A CT scan of the heart is performed to gain knowledge about cardiac or coronary anatomy.&lt;ref&gt;{{Cite web|url=https://www.nhlbi.nih.gov/health/health-topics/topics/ct|title=Cardiac CT Scan – NHLBI, NIH|website=www.nhlbi.nih.gov|archive-url=https://web.archive.org/web/20171201032800/https://www.nhlbi.nih.gov/health/health-topics/topics/ct|archive-date=2017-12-01|access-date=2017-11-22|url-status=live}}&lt;/ref&gt; Traditionally, cardiac CT scans are used to detect, diagnose, or follow up [[w:coronary artery disease|coronary artery disease]].&lt;ref name=&quot;Wichmann&quot;&gt;{{Cite web|url=https://radiopaedia.org/articles/cardiac-ct-1|title=Cardiac CT {{!}} Radiology Reference Article {{!}} Radiopaedia.org|last=Wichmann|first=Julian L.|website=radiopaedia.org|archive-url=https://web.archive.org/web/20171201040626/https://radiopaedia.org/articles/cardiac-ct-1|archive-date=2017-12-01|access-date=2017-11-22|url-status=dead}}&lt;/ref&gt; More recently CT has played a key role in the fast-evolving field of [[w:Interventional cardiology|transcatheter structural heart interventions]], more specifically in the transcatheter repair and replacement of heart valves.&lt;ref&gt;{{Cite journal|last1=Marwan|first1=Mohamed|last2=Achenbach|first2=Stephan|date=February 2016|title=Role of Cardiac CT Before Transcatheter Aortic Valve Implantation (TAVI)|journal=Current Cardiology Reports|volume=18|issue=2|pages=21|doi=10.1007/s11886-015-0696-3|issn=1534-3170|pmid=26820560|s2cid=41535442}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Moss|first1=Alastair J.|last2=Dweck|first2=Marc R.|last3=Dreisbach|first3=John G.|last4=Williams|first4=Michelle C.|last5=Mak|first5=Sze Mun|last6=Cartlidge|first6=Timothy|last7=Nicol|first7=Edward D.|last8=Morgan-Hughes|first8=Gareth J.|date=2016-11-01|title=Complementary role of cardiac CT in the assessment of aortic valve replacement dysfunction|journal=Open Heart|volume=3|issue=2|pages=e000494|doi=10.1136/openhrt-2016-000494|issn=2053-3624|pmc=5093391|pmid=27843568}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Thériault-Lauzier|first1=Pascal|last2=Spaziano|first2=Marco|last3=Vaquerizo|first3=Beatriz|last4=Buithieu|first4=Jean|last5=Martucci|first5=Giuseppe|last6=Piazza|first6=Nicolo|date=September 2015|title=Computed Tomography for Structural Heart Disease and Interventions|journal=Interventional Cardiology Review|volume=10|issue=3|pages=149–154|doi=10.15420/ICR.2015.10.03.149|issn=1756-1477|pmc=5808729|pmid=29588693}}&lt;/ref&gt; The main forms of cardiac CT scanning are:

[[w:Coronary CT angiography|Coronary CT angiography]] (CCTA): the use of CT to assess the [[w:coronary artery|coronary arteries]] of the [[w:heart|heart]]. The subject receives an [[w:intravenous injection|intravenous injection]] of [[w:radiocontrast|radiocontrast]], and then the heart is scanned using a high-speed CT scanner, allowing radiologists to assess the extent of occlusion in the coronary arteries, usually to diagnose coronary artery disease.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=eR5USB6sRU4C&amp;q=ct+angiography|title=Multidetector-Row CT Angiography|last=Passariello|first=Roberto|date=2006-03-30|publisher=Springer Science &amp; Business Media|isbn=978-3-540-26984-7|language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.radiologyinfo.org/en/info.cfm?pg=angiocoroct|title=Coronary Computed Tomography Angiography (CCTA)|last=Radiology (ACR)|first=Radiological Society of North America (RSNA) and American College of|website=www.radiologyinfo.org|language=en|access-date=2021-03-19}}&lt;/ref&gt;

[[w:Coronary CT calcium scan|Coronary CT calcium scan]]: also used for the assessment of severity of coronary artery disease.&lt;ref&gt;{{Cite journal|last=Greenland|first=Philip|date=2004-01-14|title=Coronary Artery Calcium Score Combined With Framingham Score for Risk Prediction in Asymptomatic Individuals|url=http://jama.jamanetwork.com/article.aspx?doi=10.1001/jama.291.2.210|journal=JAMA|language=en|volume=291|issue=2|pages=210|doi=10.1001/jama.291.2.210|issn=0098-7484}}&lt;/ref&gt; Specifically, it looks for calcium deposits in the coronary arteries that can narrow arteries and increase the risk of a heart attack.&lt;ref&gt;{{Cite journal|last=Gupta|first=Amit|last2=Bera|first2=Kaustav|last3=Kikano|first3=Elias|last4=Pierce|first4=Jonathan D.|last5=Gan|first5=Jonathan|last6=Rajdev|first6=Maharshi|last7=Ciancibello|first7=Leslie M.|last8=Gupta|first8=Aekta|last9=Rajagopalan|first9=Sanjay|date=2022-07|title=Coronary Artery Calcium Scoring: Current Status and Future Directions|url=http://pubs.rsna.org/doi/10.1148/rg.210122|journal=RadioGraphics|language=en|volume=42|issue=4|pages=947–967|doi=10.1148/rg.210122|issn=0271-5333}}&lt;/ref&gt; A typical coronary CT calcium scan is done without the use of radiocontrast, but it can possibly be done from contrast-enhanced images as well.&lt;ref name=&quot;van der BijlJoemai2010&quot;&gt;{{Cite journal|last1=van der Bijl|first1=Noortje|last2=Joemai|first2=Raoul M. S.|last3=Geleijns|first3=Jacob|last4=Bax|first4=Jeroen J.|last5=Schuijf|first5=Joanne D.|last6=de Roos|first6=Albert|last7=Kroft|first7=Lucia J. M.|year=2010|title=Assessment of Agatston Coronary Artery Calcium Score Using Contrast-Enhanced CT Coronary Angiography|journal=American Journal of Roentgenology|volume=195|issue=6|pages=1299–1305|doi=10.2214/AJR.09.3734|issn=0361-803X|pmid=21098187}}&lt;/ref&gt; 

To better visualize the anatomy, post-processing of the images is common.&lt;ref name=&quot;Wichmann&quot; /&gt; Most common are multiplanar reconstructions (MPR) and [[w:volume rendering|volume rendering]]. For more complex anatomies and procedures, such as heart valve interventions, a true [[w:3D reconstruction|3D reconstruction]] or a 3D print is created based on these CT images to gain a deeper understanding.&lt;ref&gt;{{Cite journal|last1=Vukicevic|first1=Marija|last2=Mosadegh|first2=Bobak|last3=Min|first3=James K.|last4=Little|first4=Stephen H.|date=February 2017|title=Cardiac 3D Printing and its Future Directions|journal=JACC: Cardiovascular Imaging|volume=10|issue=2|pages=171–184|doi=10.1016/j.jcmg.2016.12.001|issn=1876-7591|pmc=5664227|pmid=28183437}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Wang|first1=D. D.|last2=Eng|first2=M.|last3=Greenbaum|first3=A.|last4=Myers|first4=E.|last5=Forbes|first5=M.|last6=Pantelic|first6=M.|last7=Song|first7=T.|last8=Nelson|first8=C.|last9=Divine|first9=G.|year=2016|title=Innovative Mitral Valve Treatment with 3D Visualization at Henry Ford|url=http://www.materialise.com/en/blog/innovative-mitral-valve-treatment-3d-visualization-at-henry-ford|journal=JACC: Cardiovascular Imaging|volume=9|issue=11|pages=1349–1352|doi=10.1016/j.jcmg.2016.01.017|pmc=5106323|pmid=27209112|archive-url=https://web.archive.org/web/20171201043336/http://www.materialise.com/en/blog/innovative-mitral-valve-treatment-3d-visualization-at-henry-ford|archive-date=2017-12-01|access-date=2017-11-22|last10=Taylor|first10=A.|last11=Wyman|first11=J.|last12=Guerrero|first12=M.|last13=Lederman|first13=R. J.|last14=Paone|first14=G.|last15=O'Neill|first15=W.|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Wang|first1=Dee Dee|last2=Eng|first2=Marvin|last3=Greenbaum|first3=Adam|last4=Myers|first4=Eric|last5=Forbes|first5=Michael|last6=Pantelic|first6=Milan|last7=Song|first7=Thomas|last8=Nelson|first8=Christina|last9=Divine|first9=George|date=November 2016|title=Predicting LVOT Obstruction After TMVR|journal=JACC: Cardiovascular Imaging|volume=9|issue=11|pages=1349–1352|doi=10.1016/j.jcmg.2016.01.017|issn=1876-7591|pmc=5106323|pmid=27209112}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Jacobs|first1=Stephan|last2=Grunert|first2=Ronny|last3=Mohr|first3=Friedrich W.|last4=Falk|first4=Volkmar|date=February 2008|title=3D-Imaging of cardiac structures using 3D heart models for planning in heart surgery: a preliminary study|journal=Interactive Cardiovascular and Thoracic Surgery|volume=7|issue=1|pages=6–9|doi=10.1510/icvts.2007.156588|issn=1569-9285|pmid=17925319|doi-access=free}}&lt;/ref&gt;

=== Interventional ===
CT-guided interventional procedures involve minimally invasive techniques guided by computed tomography imaging. These procedures utilize detailed cross-sectional images generated by CT scans to precisely guide various interventions. Common interventions performed under CT-guidance include biopsies for diagnostic purposes, drainage of fluid-filled areas, radiofrequency ablation to destroy tumors, and procedures like vertebroplasty or kyphoplasty for stabilizing fractured vertebrae.&lt;ref&gt;{{Cite journal|last=Tsalafoutas|first=Ioannis A.|last2=Tsapaki|first2=Virginia|last3=Triantopoulou|first3=Charicleia|last4=Gorantonaki|first4=Akrivi|last5=Papailiou|first5=John|date=2007-06|title=CT-Guided Interventional Procedures without CT Fluoroscopy Assistance: Patient Effective Dose and Absorbed Dose Considerations|url=https://www.ajronline.org/doi/10.2214/AJR.06.0705|journal=American Journal of Roentgenology|language=en|volume=188|issue=6|pages=1479–1484|doi=10.2214/AJR.06.0705|issn=0361-803X}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Seong|first=Ju-Yong|last2=Kim|first2=Jin-Sung|last3=Jung|first3=Byungjoo|last4=Lee|first4=Sang-Ho|last5=Kang|first5=Ho Yeong|date=2009|title=CT-Guided Percutaneous Vertebroplasty in the Treatment of an Upper Thoracic Compression Fracture|url=https://www.kjronline.org/DOIx.php?id=10.3348/kjr.2009.10.2.185|journal=Korean Journal of Radiology|volume=10|issue=2|pages=185|doi=10.3348/kjr.2009.10.2.185|issn=1229-6929|pmc=PMC2651434|pmid=19270865}}&lt;/ref&gt; The real-time imaging provided by CT ensures accuracy in needle or catheter placement during these procedures.&lt;ref&gt;{{Cite journal|last=Lucey|first=Brian C.|last2=Varghese|first2=Jose C.|last3=Hochberg|first3=Aaron|last4=Blake|first4=Michael A.|last5=Soto|first5=Jorge A.|date=2007-05|title=CT-Guided Intervention with Low Radiation Dose: Feasibility and Experience|url=https://www.ajronline.org/doi/10.2214/AJR.06.0378|journal=American Journal of Roentgenology|language=en|volume=188|issue=5|pages=1187–1194|doi=10.2214/AJR.06.0378|issn=0361-803X}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Kliger|first=Chad|last2=Jelnin|first2=Vladimir|last3=Sharma|first3=Sonnit|last4=Panagopoulos|first4=Georgia|last5=Einhorn|first5=Bryce N.|last6=Kumar|first6=Robert|last7=Cuesta|first7=Francisco|last8=Maranan|first8=Leandro|last9=Kronzon|first9=Itzhak|date=2014-02|title=CT Angiography–Fluoroscopy Fusion Imaging for Percutaneous Transapical Access|url=https://linkinghub.elsevier.com/retrieve/pii/S1936878X1300822X|journal=JACC: Cardiovascular Imaging|language=en|volume=7|issue=2|pages=169–177|doi=10.1016/j.jcmg.2013.10.009}}&lt;/ref&gt; CT fluoroscopy proves to be a valuable clinical instrument, enhancing the efficiency of percutaneous abdominal and pelvic interventional procedures.&lt;ref&gt;{{Cite journal|last=Daly|first=Barry|last2=Templeton|first2=Philip A.|date=1999-05|title=Real-time CT Fluoroscopy: Evolution of an Interventional Tool|url=http://pubs.rsna.org/doi/10.1148/radiology.211.2.r99ma51309|journal=Radiology|language=en|volume=211|issue=2|pages=309–315|doi=10.1148/radiology.211.2.r99ma51309|issn=0033-8419}}&lt;/ref&gt; 

CT-guided biopsies encompass guided imaging techniques, employing a CT scanner to direct the needle insertion during the procedure. These guided procedures can be either diagnostic or therapeutic.&lt;ref&gt;{{Cite book|title=Percutaneous image-guided biopsy|last=Ahrar|first=Kamran|last2=Gupta|first2=Sanjay|date=2014|publisher=Springer|isbn=978-1-4614-8217-8|location=New York}} Page no. 19&lt;/ref&gt;&lt;ref&gt;{{Cite book|title=CT- and MR-guided interventions in radiology|last=Mahnken|first=Andreas H.|last2=Wilhelm|first2=Kai|last3=Ricke|first3=Jens|date=2013|publisher=Springer|isbn=978-3-642-33581-5|edition=2nd ed|location=Berlin New York}}&lt;/ref&gt; CT-guided biopsies are widely utilized for diagnosing hepatic, renal, pulmonary, bone, pancreatic, adrenal, lymphatic, and brain lesions.&lt;ref&gt;{{Cite journal|last=Hyun|first=Kwon H.A.|last2=Sachs|first2=Peter B.|last3=Haaga|first3=John R.|last4=Abdul-Karim|first4=Fadi|date=1991-04|title=CT-guided liver biopsy: An update|url=https://doi.org/10.1016/0899-7071(91)90155-O|journal=Clinical Imaging|volume=15|issue=2|pages=99–104|doi=10.1016/0899-7071(91)90155-o|issn=0899-7071}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Uppot|first=Raul N.|last2=Harisinghani|first2=Mukesh G.|last3=Gervais|first3=Debra A.|date=2010-06|title=Imaging-Guided Percutaneous Renal Biopsy: Rationale and Approach|url=https://www.ajronline.org/doi/10.2214/AJR.10.4427|journal=American Journal of Roentgenology|language=en|volume=194|issue=6|pages=1443–1449|doi=10.2214/AJR.10.4427|issn=0361-803X}}&lt;/ref&gt;

CT-guided nephrostomies demonstrate feasibility and effectiveness, particularly in instances of iatrogenic ureteral injury.&lt;ref&gt;{{Cite journal|last=Jiao|first=Dechao|last2=Li|first2=Zongming|last3=Li|first3=Zhiguo|last4=Shui|first4=Shaofeng|last5=Han|first5=Xin-wei|date=2017-08-01|title=Flat detector cone beam CT-guided nephrostomy using virtual navigation in patients with iatrogenic ureteral injury|url=https://doi.org/10.1007/s11547-017-0751-9|journal=La radiologia medica|language=en|volume=122|issue=8|pages=557–563|doi=10.1007/s11547-017-0751-9|issn=1826-6983}}&lt;/ref&gt; This technique enables precise needle placement and the detection of subtle density variations within tissues&lt;ref&gt;{{Cite journal|last=Haaga|last2=Zelch|first2=Mg|last3=Alfidi|first3=Rj|last4=Stewart|first4=Bh|last5=Daugherty|first5=Jd|date=1977-04-01|title=CT-guided antegrade pyelography and percutaneous nephrostomy|url=https://www.ajronline.org/doi/10.2214/ajr.128.4.621|journal=American Journal of Roentgenology|language=en|volume=128|issue=4|pages=621–624|doi=10.2214/ajr.128.4.621|issn=0361-803X}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Smith|first=Paul Edmund|last2=Luong|first2=Ian Thuan Hien|last3=van der Vliet|first3=Andrew Hans|date=2018-08|title=CT ‐guided nephrostomy: Re‐inventing the wheel for the occasional interventionalist|url=https://onlinelibrary.wiley.com/doi/10.1111/1754-9485.12720|journal=Journal of Medical Imaging and Radiation Oncology|language=en|volume=62|issue=4|pages=520–524|doi=10.1111/1754-9485.12720|issn=1754-9477}}&lt;/ref&gt; CT-guided percutaneous nephrostomy has proven to be efficient and safe, associated with low complication rates&lt;ref&gt;{{Cite journal|last=Egilmez|first=H.|last2=Oztoprak|first2=I.|last3=Atalar|first3=M.|last4=Cetin|first4=A.|last5=Gumus|first5=C.|last6=Gultekin|first6=Y.|last7=Bulut|first7=S.|last8=Arslan|first8=M.|last9=Solak|first9=O.|date=2007-09|title=The place of computed tomography as a guidance modality in percutaneous nephrostomy: analysis of a 10-year single-center experience|url=http://journals.sagepub.com/doi/10.1080/02841850701416528|journal=Acta Radiologica|language=en|volume=48|issue=7|pages=806–813|doi=10.1080/02841850701416528|issn=0284-1851}}&lt;/ref&gt;.

CT-guided tumor ablation involves using CT imaging for precise guidance during minimally invasive procedures to treat tumors. Techniques such as radiofrequency ablation  and microwave ablation utilize heat to destroy cancerous tissues.&lt;ref&gt;{{Cite journal|last=Engstrand|first=Jennie|last2=Toporek|first2=Grzegorz|last3=Harbut|first3=Piotr|last4=Jonas|first4=Eduard|last5=Nilsson|first5=Henrik|last6=Freedman|first6=Jacob|date=2017-01|title=Stereotactic CT-Guided Percutaneous Microwave Ablation of Liver Tumors With the Use of High-Frequency Jet Ventilation: An Accuracy and Procedural Safety Study|url=https://www.ajronline.org/doi/10.2214/AJR.15.15803|journal=American Journal of Roentgenology|language=en|volume=208|issue=1|pages=193–200|doi=10.2214/AJR.15.15803|issn=0361-803X}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Kulkarni|first=Suyash S|last2=Shetty|first2=Nitin S|last3=Polnaya|first3=Ashwin M|last4=Janu|first4=Amit|last5=Kumar|first5=Suresh|last6=Puri|first6=Ajay|last7=Gulia|first7=Ashish|last8=Rangarajan|first8=Venkatesh|date=2017-07|title=CT-guided radiofrequency ablation in osteoid osteoma: Result from a tertiary cancer centre in India|url=http://www.thieme-connect.de/DOI/DOI?10.4103/ijri.IJRI_30_17|journal=Indian Journal of Radiology and Imaging|language=en|volume=27|issue=03|pages=318–323|doi=10.4103/ijri.IJRI_30_17|issn=0971-3026}}&lt;/ref&gt; The visualization capabilities of CT enables to accurately target and ablate tumors, offering a less invasive alternative for patients with conditions such as liver tumors.&lt;ref&gt;{{Cite journal|last=Raissi|first=Driss|last2=Sanampudi|first2=Sreeja|last3=Yu|first3=Qian|last4=Winkler|first4=Michael|date=2022-01-20|title=CT-guided microwave ablation of hepatic malignancies via transpulmonary approach without ancillary techniques|url=https://clinicalimagingscience.org/ct-guided-microwave-ablation-of-hepatic-malignancies-via-transpulmonary-approach-without-ancillary-techniques/|journal=Journal of Clinical Imaging Science|language=en|volume=12|pages=2|doi=10.25259/JCIS_152_2021|issn=2156-5597|pmc=PMC8813600|pmid=35127245}}&lt;/ref&gt;

=== Vascular Imaging===
[[w:Computed tomography angiography|Computed tomography angiography]] (CTA) is a type of [[w:contrast CT|contrast CT]] to visualize the [[w:arteries|arteries]] and [[w:vein|vein]]s throughout the body.&lt;ref&gt;{{Citation |last1=McDermott |first1=M. |title=Chapter 10 – Critical care in acute ischemic stroke |date=2017-01-01 |journal=Handbook of Clinical Neurology |volume=140 |pages=153–176 |editor-last=Wijdicks |editor-first=Eelco F. M. |series=Critical Care Neurology Part I |publisher=Elsevier |language=en |doi=10.1016/b978-0-444-63600-3.00010-6 |pmid=28187798 |last2=Jacobs |first2=T. |last3=Morgenstern |first3=L. |editor2-last=Kramer |editor2-first=Andreas H.}}&lt;/ref&gt;  This ranges from arteries serving the [[w:brain|brain]] to those bringing blood to the [[w:lung|lung]]s, [[w:kidney|kidney]]s, [[w:arm|arm]]s and [[w:leg|leg]]s. An example of this type of exam is [[w:CT pulmonary angiogram|CT pulmonary angiogram]] (CTPA) used to diagnose [[w:pulmonary embolism|pulmonary embolism]] (PE). It employs computed tomography and an [[w:iodinated contrast|iodine-based contrast agent]] to obtain an image of the [[w:pulmonary artery|pulmonary arteries]].&lt;ref&gt;{{Cite journal |last1=Zeman |first1=R K |last2=Silverman |first2=P M |last3=Vieco |first3=P T |last4=Costello |first4=P |date=1995-11-01 |title=CT angiography. |journal=American Journal of Roentgenology |volume=165 |issue=5 |pages=1079–1088 |doi=10.2214/ajr.165.5.7572481 |issn=0361-803X |pmid=7572481 |doi-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last1=Ramalho |first1=Joana |url=https://books.google.com/books?id=FKdMAgAAQBAJ&amp;q=cta+is+an+imaging |title=Vascular Imaging of the Central Nervous System: Physical Principles, Clinical Applications, and Emerging Techniques |last2=Castillo |first2=Mauricio |date=2014-03-31 |publisher=John Wiley &amp; Sons |isbn=978-1-118-18875-0 |page=69 |language=en}}&lt;/ref&gt;

== Other uses ==
Industrial CT scanning (industrial computed tomography) is a process which utilizes X-ray equipment to produce 3D representations of components both externally and internally. Industrial CT scanning has been utilized in many areas of industry for internal inspection of components. Some of the key uses for CT scanning have been flaw detection, failure analysis, metrology, assembly analysis, image-based finite element methods&lt;ref&gt;{{Cite journal|last1=Evans|first1=Ll. M.|last2=Margetts|first2=L.|last3=Casalegno|first3=V.|last4=Lever|first4=L. M.|last5=Bushell|first5=J.|last6=Lowe|first6=T.|last7=Wallwork|first7=A.|last8=Young|first8=P.|last9=Lindemann|first9=A.|date=2015-05-28|title=Transient thermal finite element analysis of CFC–Cu ITER monoblock using X-ray tomography data|url=https://www.researchgate.net/publication/277338941|journal=Fusion Engineering and Design|volume=100|pages=100–111|doi=10.1016/j.fusengdes.2015.04.048|archive-url=https://web.archive.org/web/20151016091649/http://www.researchgate.net/publication/277338941_Transient_thermal_finite_element_analysis_of_CFCCu_ITER_monoblock_using_X-ray_tomography_data|archive-date=2015-10-16|doi-access=free|url-status=live}}&lt;/ref&gt; and reverse engineering applications. CT scanning is also employed in the imaging and conservation of museum artifacts.&lt;ref&gt;{{Cite journal|last=Payne, Emma Marie|year=2012|title=Imaging Techniques in Conservation|url=http://discovery.ucl.ac.uk/1443164/1/56-566-2-PB.pdf|journal=Journal of Conservation and Museum Studies|volume=10|issue=2|pages=17–29|doi=10.5334/jcms.1021201|doi-access=free}}&lt;/ref&gt;

CT scanning has also found an application in transport security (predominantly airport security) where it is currently used in a materials analysis context for explosives detection CTX (explosive-detection device)&lt;ref&gt;{{Cite book|title=Anomaly Detection and Imaging with X-Rays (ADIX) III|last1=P. Babaheidarian|last2=D. Castanon|date=2018|isbn=978-1-5106-1775-9|editor-last1=Greenberg|editor-first1=Joel A.|pages=12|chapter=Joint reconstruction and material classification in spectral CT|doi=10.1117/12.2309663|editor-last2=Gehm|editor-first2=Michael E.|editor-last3=Neifeld|editor-first3=Mark A.|editor-last4=Ashok|editor-first4=Amit|s2cid=65469251}}&lt;/ref&gt;&lt;ref name=&quot;jin12securityct&quot;&gt;{{Cite book|title=Second International Conference on Image Formation in X-Ray Computed Tomography|last1=P. Jin|last2=E. Haneda|last3=K. D. Sauer|last4=C. A. Bouman|date=June 2012|chapter=A model-based 3D multi-slice helical CT reconstruction algorithm for transportation security application|access-date=2015-04-05|chapter-url=https://engineering.purdue.edu/~bouman/publications/orig-pdf/CT-2012a.pdf|archive-url=https://web.archive.org/web/20150411000659/https://engineering.purdue.edu/~bouman/publications/orig-pdf/CT-2012a.pdf|archive-date=2015-04-11|url-status=dead}}&lt;/ref&gt;&lt;ref name=&quot;jin12securityctprior&quot;&gt;{{Cite book|title=Signals, Systems and Computers (ASILOMAR), 2012 Conference Record of the Forty Sixth Asilomar Conference on|last1=P. Jin|last2=E. Haneda|last3=C. A. Bouman|date=November 2012|publisher=IEEE|pages=613–636|chapter=Implicit Gibbs prior models for tomographic reconstruction|access-date=2015-04-05|chapter-url=https://engineering.purdue.edu/~bouman/publications/pdf/Asilomar-2012-Pengchong.pdf|archive-url=https://web.archive.org/web/20150411025559/https://engineering.purdue.edu/~bouman/publications/pdf/Asilomar-2012-Pengchong.pdf|archive-date=2015-04-11|url-status=dead}}&lt;/ref&gt;&lt;ref name=&quot;kisner13securityct&quot;&gt;{{Cite book|title=Security Technology (ICCST), 2013 47th International Carnahan Conference on|last1=S. J. Kisner|last2=P. Jin|last3=C. A. Bouman|last4=K. D. Sauer|last5=W. Garms|last6=T. Gable|last7=S. Oh|last8=M. Merzbacher|last9=S. Skatter|date=October 2013|publisher=IEEE|chapter=Innovative data weighting for iterative reconstruction in a helical CT security baggage scanner|access-date=2015-04-05|chapter-url=https://engineering.purdue.edu/~bouman/publications/pdf/iccst2013.pdf|archive-url=https://web.archive.org/web/20150410234541/https://engineering.purdue.edu/~bouman/publications/pdf/iccst2013.pdf|archive-date=2015-04-10|url-status=dead}}&lt;/ref&gt; and is also under consideration for automated baggage/parcel security scanning using computer vision based object recognition algorithms that target the detection of specific threat items based on 3D appearance (e.g. guns, knives, liquid containers).&lt;ref name=&quot;megherbi10baggage&quot;&gt;{{Cite journal|last=Megherbi|first=Najla|last2=Flitton|first2=Greg T.|last3=Breckon|first3=Toby P.|date=2010-09|title=A classifier based approach for the detection of potential threats in CT based Baggage Screening|url=https://ieeexplore.ieee.org/document/5653676/|journal=2010 IEEE International Conference on Image Processing|pages=1833–1836|doi=10.1109/ICIP.2010.5653676}}&lt;/ref&gt;&lt;ref name=&quot;megherbi12baggage&quot;&gt;{{Cite journal|last=Megherbi|first=Najla|last2=Han|first2=Jiwan|last3=Breckon|first3=Toby P.|last4=Flitton|first4=Greg T.|date=2012-09|title=A comparison of classification approaches for threat detection in CT based baggage screening|url=https://ieeexplore.ieee.org/document/6467558/|journal=2012 19th IEEE International Conference on Image Processing|pages=3109–3112|doi=10.1109/ICIP.2012.6467558}}&lt;/ref&gt;&lt;ref name=&quot;flitton13interestpoint&quot;&gt;{{Cite journal|last=Flitton|first=Greg|last2=Breckon|first2=Toby P.|last3=Megherbi|first3=Najla|date=2013-09|title=A comparison of 3D interest point descriptors with application to airport baggage object detection in complex CT imagery|url=https://doi.org/10.1016/j.patcog.2013.02.008|journal=Pattern Recognition|volume=46|issue=9|pages=2420–2436|doi=10.1016/j.patcog.2013.02.008|issn=0031-3203}}&lt;/ref&gt; 

X-ray CT is used in geological studies to quickly reveal materials inside a drill core.&lt;ref&gt;{{Cite web|url=http://www.jamstec.go.jp/chikyu/e/about/laboratory.html|title=Laboratory {{!}} About Chikyu {{!}} The Deep-sea Scientific Drilling Vessel CHIKYU|website=www.jamstec.go.jp|access-date=2019-10-24}}&lt;/ref&gt; Dense minerals such as pyrite and barite appear brighter and less dense components such as clay appear dull in CT images.&lt;ref&gt;{{Cite journal|last1=Tonai|first1=Satoshi|last2=Kubo|first2=Yusuke|last3=Tsang|first3=Man-Yin|last4=Bowden|first4=Stephen|last5=Ide|first5=Kotaro|last6=Hirose|first6=Takehiro|last7=Kamiya|first7=Nana|last8=Yamamoto|first8=Yuzuru|last9=Yang|first9=Kiho|date=2019|title=A New Method for Quality Control of Geological Cores by X-Ray Computed Tomography: Application in IODP Expedition 370|journal=Frontiers in Earth Science|language=English|volume=7|doi=10.3389/feart.2019.00117|issn=2296-6463|doi-access=free|last10=Yamada|first10=Yasuhiro|last11=Morono|first11=Yuki|s2cid=171394807}}&lt;/ref&gt;

X-ray CT and micro-CT can also be used for the conservation and preservation of objects of cultural heritage. For many fragile objects, direct research and observation can be damaging and can degrade the object over time. Using CT scans, conservators and researchers are able to determine the material composition of the objects they are exploring, such as the position of ink along the layers of a scroll, without any additional harm. After scanning these objects, computational methods can be employed to examine the insides of these objects.&lt;ref&gt;{{Cite journal|last1=Seales|first1=W. B.|last2=Parker|first2=C. S.|last3=Segal|first3=M.|last4=Tov|first4=E.|last5=Shor|first5=P.|last6=Porath|first6=Y.|year=2016|title=From damage to discovery via virtual unwrapping: Reading the scroll from En-Gedi|journal=Science Advances|volume=2|issue=9|pages=e1601247|bibcode=2016SciA....2E1247S|doi=10.1126/sciadv.1601247|issn=2375-2548|pmc=5031465|pmid=27679821}}&lt;/ref&gt; Micro-CT has also proved useful for analyzing more recent artifacts such as still-sealed historic correspondence that employed the technique of letterlocking (complex folding and cuts) that provided a &quot;tamper-evident locking mechanism&quot;.&lt;ref&gt;{{Cite web|url=https://www.wsj.com/articles/a-letter-sealed-for-centuries-has-been-readwithout-even-opening-it-11614679203|title=A Letter Sealed for Centuries Has Been Read—Without Even Opening It|last=Castellanos|first=Sara|date=2 March 2021|website=The Wall Street Journal|access-date=2 March 2021}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Dambrogio|first1=Jana|last2=Ghassaei|first2=Amanda|last3=Staraza Smith|first3=Daniel|last4=Jackson|first4=Holly|last5=Demaine|first5=Martin L.|date=2 March 2021|title=Unlocking history through automated virtual unfolding of sealed documents imaged by X-ray microtomography|journal=Nature Communications|volume=12|issue=1|page=1184|bibcode=2021NatCo..12.1184D|doi=10.1038/s41467-021-21326-w|pmc=7925573|pmid=33654094}}&lt;/ref&gt;

== Procedure ==
Before starting the procedure, the patient preparation is necessary to ensure optimal scan quality and safety. This preparation includes a thorough examination of the patient's medical history to identify any potential contraindications. The patient is briefed about the procedure, and informed written consent is obtained from the patient on family member.[[File:CT ScoutView.jpg|thumb|206x206px|Topogram ]]The specific preparation measures vary depending on the type of scan and the targeted organ. For abdominal or pelvic CT scans, fasting is essential to minimize interference from bowel gas and enhance the visualization of organs. Pre-scan instructions are also influenced by the use of contrast material, with some patients advised to refrain from certain medications, especially those affecting kidney function.

Patients undergoing CT scans may experience anxiety, either due to the unfamiliar environment or, in some cases, claustrophobia.&lt;ref&gt;{{Cite journal|last=Heyer|first=Christoph M.|last2=Thüring|first2=Johannes|last3=Lemburg|first3=Stefan P.|last4=Kreddig|first4=Nina|last5=Hasenbring|first5=Monika|last6=Dohna|first6=Martha|last7=Nicolas|first7=Volkmar|date=2015-01|title=Anxiety of Patients Undergoing CT Imaging—An Underestimated Problem?|url=https://linkinghub.elsevier.com/retrieve/pii/S1076633214002980|journal=Academic Radiology|language=en|volume=22|issue=1|pages=105–112|doi=10.1016/j.acra.2014.07.014}}&lt;/ref&gt; Consequently, maintaining stillness during the examination can be challenging for them. In such cases, commonly in children, a sedative can be employed to alleviate the patient's anxiety and ensure a smoother scanning process.&lt;ref&gt;{{Cite journal|last=Keeter|first=S|last2=Benator|first2=R M|last3=Weinberg|first3=S M|last4=Hartenberg|first4=M A|date=1990-06|title=Sedation in pediatric CT: national survey of current practice.|url=http://pubs.rsna.org/doi/10.1148/radiology.175.3.2343126|journal=Radiology|language=en|volume=175|issue=3|pages=745–752|doi=10.1148/radiology.175.3.2343126|issn=0033-8419}}&lt;/ref&gt;

Before the actual scan, a topogram, also known as a scout image or localizer image, is taken which is a low-dose, low-resolution radiographic image. This initial image helps to plan the coverage and orientation of the subsequent CT scan. The topogram provides a preliminary overview of the area to be imaged, allowing technologist to plan the scan.&lt;ref&gt;{{Cite journal|last=Li|first=Baojun|last2=Behrman|first2=Richard H.|last3=Norbash|first3=Alexander M.|date=2012-06|title=Comparison of topogram‐based body size indices for CT dose consideration and scan protocol optimization|url=https://aapm.onlinelibrary.wiley.com/doi/10.1118/1.4718569|journal=Medical Physics|language=en|volume=39|issue=6Part1|pages=3456–3465|doi=10.1118/1.4718569|issn=0094-2405}}&lt;/ref&gt; Since, Topograms have a larger field of view than main scan, they can also play a role in revealing significant findings outside the scan field of view.&lt;ref&gt;{{Cite journal|last=Lee|first=Matthew H.|last2=Lubner|first2=Meghan G.|last3=Mellnick|first3=Vincent M.|last4=Menias|first4=Christine O.|last5=Bhalla|first5=Sanjeev|last6=Pickhardt|first6=Perry J.|date=2021-10|title=The CT scout view: complementary value added to abdominal CT interpretation|url=https://link.springer.com/10.1007/s00261-021-03135-3|journal=Abdominal Radiology|language=en|volume=46|issue=10|pages=5021–5036|doi=10.1007/s00261-021-03135-3|issn=2366-004X}}&lt;/ref&gt;

=== Non-contrast CT ===
[[File:CT ABDOMEN.jpg|thumb|194x194px|Non Contrast CT Abdomen]]
CT procedure in which contrast media is not used is often called as Non-Contrast CT (NCCT) or plain CT. This procedure is employed when there is already a sufficient contrast distinction in the target tissues, rendering the resulting image diagnostically significant. The process involves acquiring a topogram, followed by scanning the region of interest and reconstructing the data, marking the conclusion of the procedure. The non-contrast CT scans are rapid, less hazardous, and cost-effective procedures. Non-contrast CT head scans play a pivotal role in the identification of various conditions, encompassing traumatic hemorrhages, subdural hematomas, cerebral edema, fractures, and in detecting  foreign bodies, such as tempered glass, which might be overlooked&lt;ref&gt;{{Cite journal|last=Faheem|first=Mohd|last2=Kumar|first2=Raj|last3=Jaiswal|first3=Manish|last4=Ansari|first4=Mohammad Ahmed|last5=Saba|first5=Noor us|date=2019-12|title=“Mosaic Pattern” Foreign Bodies in Computed Tomography of the Head: A Specific Sign to Detect Tempered Glass-Related Head Injury|url=http://www.thieme-connect.de/DOI/DOI?10.1055/s-0039-1700302|journal=Indian Journal of Neurosurgery|language=en|volume=08|issue=03|pages=193–195|doi=10.1055/s-0039-1700302|issn=2277-954X}}&lt;/ref&gt;.   

=== Contrast CT ===
[[File:Normal contrast enhanced abdominal CT.jpg|thumb|Normal contrast enhanced abdominal CT.]]
Contrast CT or CECT procedures invole the use of a contrast medium for better visualization. Contrast media, also known as contrast agents, are substances used in imaging to improve the visibility of internal structures or fluids during diagnostic procedures. These agents enhance the differentiation between various tissues, and between normal and abnormal tissues allowing for clearer and more detailed imaging.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=xb-xLHTqOi0C&amp;q=contrast+in+ct|title=Fundamentals of Body CT|last1=Webb|first1=Wayne Richard|last2=Brant|first2=William E.|last3=Major|first3=Nancy M.|date=2006-01-01|publisher=Elsevier Health Sciences|isbn=978-1-4160-0030-3|page=168|language=en}}&lt;/ref&gt; Contrast agents employed in CT imaging also know as Radio contrasts are generally categorized into Positive, Negative, and Neutral contrast agents. Positive contrast agents increase the x-ray attenuation and negative contrast agents reduce x-ray attenuation. Neutral contrast media, on the other hand, do not alter attenuation but are employed to enhance distention.&lt;ref&gt;{{Cite journal|last=Callahan|first=Michael J.|last2=Talmadge|first2=Jennifer M.|last3=MacDougall|first3=Robert|last4=Buonomo|first4=Carlo|last5=Taylor|first5=George A.|date=2016-05|title=The Use of Enteric Contrast Media for Diagnostic CT, MRI, and Ultrasound in Infants and Children: A Practical Approach|url=https://www.ajronline.org/doi/10.2214/AJR.15.15437|journal=American Journal of Roentgenology|language=en|volume=206|issue=5|pages=973–979|doi=10.2214/AJR.15.15437|issn=0361-803X}}&lt;/ref&gt;    [[File:CT Contrast classification.png|thumb|CT Contrast classification|231x231px|left]]
Positive contrast agents can be categorized into Iodinated, Oily, or Barium sulfate contrast agents based on their composition. The most prevalent among them are Iodinated contrast agents, which are based on Iodine. These are further classified into Ionic contrast media and non-ionic contrast media.&lt;ref&gt;{{Cite journal|last=Baerlocher|first=M. O.|last2=Asch|first2=M.|last3=Myers|first3=A.|date=2010-04-20|title=The use of contrast media|url=http://www.cmaj.ca/cgi/doi/10.1503/cmaj.090118|journal=Canadian Medical Association Journal|language=en|volume=182|issue=7|pages=697–697|doi=10.1503/cmaj.090118|issn=0820-3946|pmc=PMC2855918|pmid=20231343}}&lt;/ref&gt; The ionic iodinated contrast is further divided into Ionic monomers and Ionic dimers, &amp; the  non ionic iodinated contrast media is divided into non-ionic monomers and non-ionic dimers. Ionic monomers constitute a type of high-osmolar contrast media with an Iodine-to-particle ratio of 3:2, while Ionic dimers and Non-ionic monomers share an Iodine-to-particle ratio of 3:1. Non-ionic dimers, on the other hand, have a ratio of 6:1. The decrease in osmolarity is associated with the increase in the viscosity.&lt;ref&gt;{{Cite journal|last=Bottinor|first=Wendy|last2=Polkampally|first2=Pritam|last3=Jovin|first3=Ion|date=2013-08-16|title=Adverse Reactions to Iodinated Contrast Media|url=http://www.thieme-connect.de/DOI/DOI?10.1055/s-0033-1348885|journal=International Journal of Angiology|language=en|volume=22|issue=03|pages=149–154|doi=10.1055/s-0033-1348885|issn=1061-1711}}&lt;/ref&gt;  

Barium based contrast are used in the imaging of gastrointestinal tract. These contrast agents help outline the contours of the GI organs, such as the stomach and intestines, making them more visible on CT scans. Barium provides a high degree of contrast due to high x ray attenuation. It can also be used for double contrast studies eg in case of barium enema. It is water insoluble and is not absorbed by the gut.        

Mainly Intravenous iodinated contrast media are used as positive contrast agents is used in CT, with some procedures Oral contrast and Negative contrast media can also be used. The use of contrast media mandates a thorough review of the patient's medical history and allergies, &amp;  assessment of renal function. Explicit written consent is imperative before administering the contrast material, delivered through an intravenous line in the arm or hand at a controlled rate via hand injection or a pressure injector. Upon injecting, the scan is initiated with precise timing. This temporal coordination is of paramount importance for observing distinct levels of enhancement throughout the scanning process.    

Contrast phases refer to distinct stages in the enhancement of blood vessels or tissue after the administration of a intravenous contrast agent during a CT procedure. During a contrast-enhanced CT scan, various contrast phases delineate the dynamic enhancement of blood vessels. The early arterial phase manifests 15-20 seconds after injection of contrast. Following this, the late arterial phase transpires 30-40 seconds post-injection. Subsequently, the portal venous phase unfolds 70-90 seconds after injection. The nephrogenic phase starts at 100-120 seconds post-injection. The excretory phase also called as washout phase occurs at 5-10 minutes after injection. 

==== Contrast injection techniques ====
Test bolus is a small, preliminary injection of contrast agent given to a patient before the actual CT scan. The purpose of the test bolus is to determine the optimal timing for the contrast-enhanced scan and in the mean time also assess the integrity of venous access before administering the complete bolus of contrast medium.&lt;ref&gt;{{Cite journal|last=Bae|first=Kyongtae T.|date=2003-06|title=Peak Contrast Enhancement in CT and MR Angiography: When Does It Occur and Why? Pharmacokinetic Study in a Porcine Model|url=http://pubs.rsna.org/doi/10.1148/radiol.2273020102|journal=Radiology|language=en|volume=227|issue=3|pages=809–816|doi=10.1148/radiol.2273020102|issn=0033-8419}}&lt;/ref&gt;

Bolus tracking is a technique employed in CECT to monitor the concentration of contrast material within a designated region of interest, typically a blood vessel. A region of interest (ROI) is typically positioned just before the target organ, The scanning process starts when the contrast concentration attains a predefined HU level, ensuring optimal filling of blood vessels with contrast. This approach aids in acquiring images at the point of peak enhancements.&lt;ref&gt;{{Cite journal|last=Nebelung|first=Heiner|last2=Brauer|first2=Thomas|last3=Seppelt|first3=Danilo|last4=Hoffmann|first4=Ralf-Thorsten|last5=Platzek|first5=Ivan|date=2021-02|title=Coronary computed tomography angiography (CCTA): effect of bolus-tracking ROI positioning on image quality|url=https://link.springer.com/10.1007/s00330-020-07131-x|journal=European Radiology|language=en|volume=31|issue=2|pages=1110–1118|doi=10.1007/s00330-020-07131-x|issn=0938-7994|pmc=PMC7813743|pmid=32809163}}&lt;/ref&gt;

== Mechanism ==
[[File:ct-internals.jpg|thumb|right|CT scanner with cover removed to show internal components. Legend:
&lt;br /&gt;T: X-ray tube
&lt;br /&gt;D: X-ray detectors
&lt;br /&gt;X: X-ray beam
&lt;br /&gt;R: Gantry rotation]]
[[File:Sinogram and sample image of computed tomography of the jaw.jpg|thumb|Left image is a ''sinogram'' which is a graphic representation of the raw data obtained from a CT scan. At right is an image sample derived from the raw data.&lt;ref&gt;{{Cite journal|last1=Jun|first1=Kyungtaek|last2=Yoon|first2=Seokhwan|year=2017|title=Alignment Solution for CT Image Reconstruction using Fixed Point and Virtual Rotation Axis|journal=Scientific Reports|volume=7|pages=41218|arxiv=1605.04833|bibcode=2017NatSR...741218J|doi=10.1038/srep41218|issn=2045-2322|pmc=5264594|pmid=28120881}}&lt;/ref&gt;]]
Computed tomography scanner operates by using an X-ray tube that generates X-rays and rotates around the patient; [[w:X-ray detector|X-ray detector]]s are positioned on the opposite side of the the X-ray source.&lt;ref&gt;{{Cite web|url=https://www.nibib.nih.gov/science-education/science-topics/computed-tomography-ct|title=Computed Tomography (CT)|website=www.nibib.nih.gov|access-date=2021-03-20}}&lt;/ref&gt; As the X-rays pass through the patient, they are attenuated by various tissues according to their density.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=nPisjRy4LNAC&amp;pg=PA3|title=Radiation Exposure and Image Quality in X-Ray Diagnostic Radiology: Physical Principles and Clinical Applications|last1=Aichinger|first1=Horst|last2=Dierker|first2=Joachim|last3=Joite-Barfuß|first3=Sigrid|last4=Säbel|first4=Manfred|date=2011-10-25|publisher=Springer Science &amp; Business Media|isbn=978-3-642-11241-6|pages=5|language=en}}&lt;/ref&gt; Tissues with higher density attenuate more x-ray photons while tissues with low density attenuate less, this attenuation data is acquired by the detectors around the patient.  A visual representation of the raw data obtained is called a sinogram, yet it is not sufficient for interpretation.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=ylcfAQAAMAAJ&amp;q=A+set+of+many+such+projections+under+different+angles+organized+in+2D+is+called+sinogram|title=Statistical Image Reconstruction Algorithms Using Paraboloidal Surrogates for PET Transmission Scans|last=Erdoğan|first=Hakan|date=1999|publisher=University of Michigan|isbn=978-0-599-63374-2|language=en}}&lt;/ref&gt; The term [[wikt:sinogram|sinogram]] was introduced by Paul Edholm and Bertil Jacobson in 1975.&lt;ref&gt;{{Cite journal|last1=Edholm|first1=Paul|last2=Gabor|first2=Herman|date=December 1987|title=Linograms in Image Reconstruction from Projections|journal=IEEE Transactions on Medical Imaging|volume=MI-6|issue=4|pages=301–7|doi=10.1109/tmi.1987.4307847|pmid=18244038|s2cid=20832295}}&lt;/ref&gt; Once the scan data has been acquired, it is then processed using a form of [[w:tomographic reconstruction|tomographic reconstruction]], which produces a series of cross-sectional images.&lt;ref&gt;{{Cite web|url=https://radiologykey.com/ct-image-reconstruction-basics/|title=CT Image Reconstruction Basics|last=Themes|first=U. F. O.|date=2018-10-07|website=Radiology Key|language=en-US|access-date=2021-03-20}}&lt;/ref&gt; These cross-sectional images are made up of small units of pixels or voxels.&lt;ref name=&quot;Cardiovascular Computed Tomography&quot;&gt;{{Cite book|url=https://books.google.com/books?id=SarDDwAAQBAJ&amp;q=ct+images+are+made+of+pixels&amp;pg=PA134|title=Cardiovascular Computed Tomography|last=Stirrup|first=James|date=2020-01-02|publisher=Oxford University Press|isbn=978-0-19-880927-2|language=en}}&lt;/ref&gt;

A [[w:pixel|pixel]] is a two dimensional unit based on the matrix size and the field of view. [[w:Pixel|Pixel]]s in an image obtained by CT scanning are displayed in terms of relative [[w:radiodensity|radiodensity]]. The pixel itself is displayed according to the mean [[w:attenuation|attenuation]] of the tissue(s) that it corresponds to on a scale from +3,071 (most attenuating) to −1,024 (least attenuating) on the [[w:Hounsfield scale|Hounsfield scale]]. When the CT slice thickness is also factored in, the unit is known as a [[w:voxel|voxel]], which is a three-dimensional unit.&lt;ref name=&quot;Cardiovascular Computed Tomography&quot; /&gt;

Water has an attenuation of 0 [[w:Hounsfield units|Hounsfield units]] (HU), while air is −1,000&amp;nbsp;HU, cancellous bone is typically +400&amp;nbsp;HU, and cranial bone can reach 2,000&amp;nbsp;HU or more (os temporale) and can cause [[w:artifact (error)#Medical imaging|artifacts]]. The attenuation of metallic implants depends on the atomic number of the element used: Titanium usually has an amount of +1000&amp;nbsp;HU, iron steel can completely block the X-ray and is, therefore, responsible for well-known line-artifacts in computed tomograms. Artifacts are caused by abrupt transitions between low- and high-density materials, which results in data values that exceed the dynamic range of the processing electronics. 

Initially, the CT scanners generated images in only [[w:transverse plane|transverse]] (axial) [[w:anatomical plane|anatomical plane]], perpendicular to the long axis of the body. Modern scanners allow the scan data to be reformatted as images in other [[w:Plane (geometry)|planes]]. [[w:Geometry processing|Digital geometry processing]] can generate a [[w:three-dimensional space|three-dimensional]] image of an object inside the body from a series of two-dimensional [[w:radiography|radiographic]] images taken by [[w:rotation around a fixed axis|rotation around a fixed axis]].&lt;ref name=&quot;ref1&quot;&gt;{{Cite book|url=https://books.google.com/books?id=JX__lLLXFHkC&amp;q=ct+can+have+a+number+of+artifacts&amp;pg=PA167|title=Computed Tomography: Principles, Design, Artifacts, and Recent Advances|last=Hsieh|first=Jiang|date=2003|publisher=SPIE Press|isbn=978-0-8194-4425-7|pages=167|language=en}}&lt;/ref&gt; These cross-sectional images are widely used for medical [[w:diagnosis|diagnosis]] and [[w:therapy|therapy]].&lt;ref name=&quot;urlcomputed tomography – Definition from the Merriam-Webster Online Dictionary&quot;&gt;{{Cite web|url=http://www.merriam-webster.com/dictionary/computed+tomography|title=computed tomography – Definition from the Merriam-Webster Online Dictionary|archive-url=https://web.archive.org/web/20110919202302/http://www.merriam-webster.com/dictionary/computed+tomography|archive-date=19 September 2011|access-date=18 August 2009|url-status=live}}&lt;/ref&gt;

== Interpretation of results ==
=== Presentation ===
[[File:CT presentation as thin slice, projection and volume rendering.jpg|thumb|224x224px|&lt;small&gt;Types of presentations of CT scans:&lt;/small&gt;]]

The result of a CT scan is a volume of [[w:voxel|voxel]]s, which may be presented to a human observer by various methods, which broadly fit into the following categories:
*Slices (of varying thickness). Thin slice is generally regarded as planes representing a thickness of less than 3 [[w:Millimetre|mm]].&lt;ref name=&quot;Goldman2008&quot;&gt;{{Cite journal |last=Goldman |first=L. W. |year=2008 |title=Principles of CT: Multislice CT |journal=Journal of Nuclear Medicine Technology |volume=36 |issue=2 |pages=57–68 |doi=10.2967/jnmt.107.044826 |issn=0091-4916 |pmid=18483143 |doi-access=free}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{Cite journal |last1=Reis |first1=Eduardo Pontes |last2=Nascimento |first2=Felipe |last3=Aranha |first3=Mateus |last4=Mainetti Secol |first4=Fernando |last5=Machado |first5=Birajara |last6=Felix |first6=Marcelo |last7=Stein |first7=Anouk |last8=Amaro |first8=Edson |date=29 July 2020 |title=Brain Hemorrhage Extended (BHX): Bounding box extrapolation from thick to thin slice CT images v1.1 |journal=PhysioNet |language=en |volume=101 |issue=23 |pages=215–220 |doi=10.13026/9cft-hg92}}&lt;/ref&gt; Thick slice is generally regarded as planes representing a thickness between 3&amp;nbsp;mm and 5&amp;nbsp;mm.&lt;ref name=&quot;:2&quot; /&gt;&lt;ref&gt;{{Cite journal |last1=Park |first1=S. |last2=Chu |first2=L.C. |last3=Hruban |first3=R.H. |last4=Vogelstein |first4=B. |last5=Kinzler |first5=K.W. |last6=Yuille |first6=A.L. |last7=Fouladi |first7=D.F. |last8=Shayesteh |first8=S. |last9=Ghandili |first9=S. |last10=Wolfgang |first10=C.L. |last11=Burkhart |first11=R. |last12=He |first12=J. |last13=Fishman |first13=E.K. |last14=Kawamoto |first14=S. |date=2020-09-01 |title=Differentiating autoimmune pancreatitis from pancreatic ductal adenocarcinoma with CT radiomics features |journal=Diagnostic and Interventional Imaging |language=en |volume=101 |issue=9 |pages=555–564 |doi=10.1016/j.diii.2020.03.002 |issn=2211-5684 |pmid=32278586 |s2cid=215751181}}&lt;/ref&gt;
*Projection, including [[w:maximum intensity projection|maximum intensity projection]]&lt;ref name=&quot;FishmanNey2006&quot;&gt;{{Cite journal |last1=Fishman |first1=Elliot K. |author-link=Elliot K. Fishman |last2=Ney |first2=Derek R. |last3=Heath |first3=David G. |last4=Corl |first4=Frank M. |last5=Horton |first5=Karen M. |last6=Johnson |first6=Pamela T. |year=2006 |title=Volume Rendering versus Maximum Intensity Projection in CT Angiography: What Works Best, When, and Why |journal=RadioGraphics |volume=26 |issue=3 |pages=905–922 |doi=10.1148/rg.263055186 |issn=0271-5333 |pmid=16702462 |doi-access=free}}&lt;/ref&gt; and ''average intensity projection''
*[[w:Volume rendering|Volume rendering]] (VR)&lt;ref name=&quot;FishmanNey2006&quot; /&gt;

Technically, all volume renderings become projections when viewed on a [[w:Display device#Full-area 2-dimensional displays|2-dimensional display]], making the distinction between projections and volume renderings a bit vague. The epitomes of volume rendering models feature a mix of for example coloring and shading in order to create realistic and observable representations.&lt;ref name=&quot;SilversteinParsad2008&quot;&gt;{{Cite journal |last1=Silverstein |first1=Jonathan C. |last2=Parsad |first2=Nigel M. |last3=Tsirline |first3=Victor |year=2008 |title=Automatic perceptual color map generation for realistic volume visualization |journal=Journal of Biomedical Informatics |volume=41 |issue=6 |pages=927–935 |doi=10.1016/j.jbi.2008.02.008 |issn=1532-0464 |pmc=2651027 |pmid=18430609}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last=Kobbelt |first=Leif |url=https://books.google.com/books?id=zndnSzkfkXwC |title=Vision, Modeling, and Visualization 2006: Proceedings, November 22-24, 2006, Aachen, Germany |date=2006 |publisher=IOS Press |isbn=978-3-89838-081-2 |pages=185 |language=en}}&lt;/ref&gt;

Two-dimensional CT images are conventionally rendered so that the view is as though looking up at it from the patient's feet. Hence, the left side of the image is to the patient's right and vice versa, while anterior in the image also is the patient's anterior and vice versa. This left-right interchange corresponds to the view that physicians generally have in reality when positioned in front of patients.&lt;ref&gt;{{Cite journal |last1=Schmidt |first1=Derek |last2=Odland |first2=Rick |date=September 2004 |title=Mirror-Image Reversal of Coronal Computed Tomography Scans |journal=The Laryngoscope |language=en |volume=114 |issue=9 |pages=1562–1565 |doi=10.1097/00005537-200409000-00011 |issn=0023-852X |pmid=15475782 |s2cid=22320649}}&lt;/ref&gt;

==== Grayscale ====
[[w:Pixel|Pixel]]s in an image obtained by CT scanning are displayed in terms of relative [[w:radiodensity|radiodensity]]. The pixel itself is displayed according to the mean [[w:attenuation|attenuation]] of the tissue(s) that it corresponds to on a scale from +3,071 (most attenuating) to −1,024 (least attenuating) on the [[w:Hounsfield scale|Hounsfield scale]]. A [[w:pixel|pixel]] is a two dimensional unit based on the matrix size and the field of view. When the CT slice thickness is also factored in, the unit is known as a [[w:voxel|voxel]], which is a three-dimensional unit.&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=63xxDwAAQBAJ |title=Brant and Helms' fundamentals of diagnostic radiology |date=2018-07-19 |publisher=Lippincott Williams &amp; Wilkins |isbn=978-1-4963-6738-9 |edition=Fifth |pages=1600 |access-date=24 January 2019}}&lt;/ref&gt; Water has an attenuation of 0 [[w:Hounsfield units|Hounsfield units]] (HU), while air is −1,000&amp;nbsp;HU, cancellous bone is typically +400&amp;nbsp;HU, and cranial bone can reach 2,000&amp;nbsp;HU.&lt;ref&gt;{{Cite book |title=Brain mapping: the methods |date=2002 |publisher=Academic Press |isbn=0-12-693019-8 |editor-last=Arthur W. Toga |edition=2nd |location=Amsterdam |oclc=52594824 |editor-last2=John C. Mazziotta}}&lt;/ref&gt; The attenuation of metallic implants depends on the atomic number of the element used: Titanium usually has an amount of +1000&amp;nbsp;HU, iron steel can completely block the X-ray and is, therefore, responsible for well-known line-artifacts in computed tomograms. Artifacts are caused by abrupt transitions between low- and high-density materials, which results in data values that exceed the dynamic range of the processing electronics.&lt;ref name=&quot;...&quot;&gt;{{Cite book |last1=Jerrold T. Bushberg |title=The essential physics of medical imaging |last2=J. Anthony Seibert |last3=Edwin M. Leidholdt |last4=John M. Boone |date=2002 |publisher=Lippincott Williams &amp; Wilkins |isbn=0-683-30118-7 |edition=2nd |location=Philadelphia |page=358 |oclc=47177732}}&lt;/ref&gt;

==== Windowing ====
CT data sets have a very high [[w:dynamic range|dynamic range]] which must be reduced for display or printing. This is typically done via a process of &quot;windowing&quot;, which maps a range (the &quot;window&quot;) of pixel values to a grayscale ramp. For example, CT images of the brain are commonly viewed with a window extending from 0 HU to 80 HU. Pixel values of 0 and lower, are displayed as black; values of 80 and higher are displayed as white; values within the window are displayed as a grey intensity proportional to position within the window.&lt;ref&gt;{{Cite journal |last1=Kamalian |first1=Shervin |last2=Lev |first2=Michael H. |last3=Gupta |first3=Rajiv |date=2016-01-01 |title=Computed tomography imaging and angiography – principles |journal=Handbook of Clinical Neurology |language=en |volume=135 |pages=3–20 |doi=10.1016/B978-0-444-53485-9.00001-5 |isbn=978-0-444-53485-9 |issn=0072-9752 |pmid=27432657}}&lt;/ref&gt; The window used for display must be matched to the X-ray density of the object of interest, in order to optimize the visible detail.&lt;ref&gt;{{Cite book |last=Stirrup |first=James |url=https://books.google.com/books?id=SarDDwAAQBAJ&amp;q=windowing+in+ct&amp;pg=PA136 |title=Cardiovascular Computed Tomography |date=2020-01-02 |publisher=Oxford University Press |isbn=978-0-19-880927-2 |page=136 |language=en}}&lt;/ref&gt; Window width and window level parameters are used to control the windowing of a scan.&lt;ref&gt;{{Cite book |last=Carroll |first=Quinn B. |url=https://books.google.com/books?id=iTwYI5rzeRMC&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA512&amp;dq=window+width+and+window+level&amp;hl=en |title=Practical Radiographic Imaging |date=2007 |publisher=Charles C Thomas Publisher |isbn=978-0-398-08511-7|page=512 |language=en}}&lt;/ref&gt;

==== Multiplanar reconstruction and projections ====
[[File:Ct-workstation-neck.jpg|thumb|Image showing one volume rendering (VR) and multiplanar view of three thin slices in the [[w:axial plane|axial]] (upper right), [[w:sagittal plane|sagittal]] (lower left), and [[w:coronal plane|coronal plane]]s (lower right)|200x200px]]
Multiplanar reconstruction also known as MPR is the process of converting data from one [[w:anatomical plane|anatomical plane]] (usually [[w:Transverse plane|transverse]]) to other planes. It can be used for thin slices as well as projections. Multiplanar reconstruction is possible as present  CT scanners provide almost [[w:isotropy|isotropic]] resolution.&lt;ref name=&quot;ref3&quot;&gt;{{Cite book |last1=Udupa |first1=Jayaram K. |url=https://books.google.com/books?id=aR6PHYluq4oC&amp;q=3D+Imaging+in+Medicine%2C+2nd+Edition |title=3D Imaging in Medicine, Second Edition |last2=Herman |first2=Gabor T. |date=1999-09-28 |publisher=CRC Press |isbn=978-0-8493-3179-4 |language=en}}&lt;/ref&gt; MPR is used almost in every scan. The spine is frequently examined with it.&lt;ref&gt;{{Cite journal |last1=Krupski |first1=Witold |last2=Kurys-Denis |first2=Ewa |last3=Matuszewski |first3=Łukasz |last4=Plezia |first4=Bogusław |date=2007-06-30 |title=Use of multi-planar reconstruction (MPR) and 3-dimentional &amp;#91;sic&amp;#93; (3D) CT to assess stability criteria in C2 vertebral fractures |url=http://www.jpccr.eu/Use-of-multi-planar-reconstruction-MPR-and-3-dimentional-3D-CT-to-assess-stability,71238,0,2.html |journal=Journal of Pre-Clinical and Clinical Research |language=english |volume=1 |issue=1 |pages=80–83 |issn=1898-2395}}&lt;/ref&gt; An image of the spine in axial plane can only show one vertebral bone at a time and cannot show its relation with other vertebral bones. By reformatting the data in other planes, visualization of the relative position can be achieved in sagittal and coronal plane.&lt;ref&gt;{{Cite journal |last=Tins |first=Bernhard |date=2010-10-21 |title=Technical aspects of CT imaging of the spine |journal=Insights into Imaging |volume=1 |issue=5–6 |pages=349–359 |doi=10.1007/s13244-010-0047-2 |issn=1869-4101 |pmc=3259341 |pmid=22347928}}&lt;/ref&gt; New software allows the reconstruction of data in non-orthogonal (oblique) planes, which help in the visualization of organs which are not in orthogonal planes.&lt;ref&gt;{{Cite book |last1=Wolfson |first1=Nikolaj |url=https://books.google.com/books?id=8Y5FDAAAQBAJ&amp;q=Modern+software+allows+reconstruction+in+non-orthogonal&amp;pg=PA373 |title=Orthopedics in Disasters: Orthopedic Injuries in Natural Disasters and Mass Casualty Events |last2=Lerner |first2=Alexander |last3=Roshal |first3=Leonid |date=2016-05-30 |publisher=Springer |isbn=978-3-662-48950-5 |language=en}}&lt;/ref&gt; It is better suited for visualization of the anatomical structure of the bronchi as they do not lie orthogonal to the direction of the scan.&lt;ref&gt;{{Cite journal |last1=Laroia |first1=Archana T |last2=Thompson |first2=Brad H |last3=Laroia |first3=Sandeep T |last4=van Beek |first4=Edwin JR |date=2010-07-28 |title=Modern imaging of the tracheo-bronchial tree |journal=World Journal of Radiology |volume=2 |issue=7 |pages=237–248 |doi=10.4329/wjr.v2.i7.237 |issn=1949-8470 |pmc=2998855 |pmid=21160663}}&lt;/ref&gt;

Curved-plane reconstruction is performed mainly for the evaluation of vessels. This type of reconstruction helps to straighten the bends in a vessel, thereby helping to visualize a whole vessel in a single image or in multiple images. After a vessel has been &quot;straightened&quot;, measurements such as cross-sectional area and length can be made. This is helpful in preoperative assessment of a surgical procedure.&lt;ref&gt;{{Cite journal |last1=Gong |first1=Jing-Shan |last2=Xu |first2=Jian-Min |date=2004-07-01 |title=Role of curved planar reformations using multidetector spiral CT in diagnosis of pancreatic and peripancreatic diseases |journal=World Journal of Gastroenterology |volume=10 |issue=13 |pages=1943–1947 |doi=10.3748/wjg.v10.i13.1943 |issn=1007-9327 |pmc=4572236 |pmid=15222042}}&lt;/ref&gt;

==== Maximum Intensity Projection (MIP) ====
Maximum Intensity Projection is a scan visualization technique which is used to highlight the highest intensity voxels along a specific projection path. In MIP, each pixel in the final image represents the maximum intensity encountered along a ray traced through the volume data. This method is particularly useful in angiography and vascular imaging, where it enhances the visualization of blood vessels by emphasizing the contrast between vessels and surrounding tissues. MIP projections are valuable for detecting abnormalities and assessing the vascular anatomy with greater clarity.&lt;ref&gt;{{Cite journal|last=Mroz|first=Lukas|last2=König|first2=Andreas|last3=Gröller|first3=Eduard|date=2000-06-01|title=Maximum intensity projection at warp speed|url=https://www.sciencedirect.com/science/article/pii/S0097849300000303|journal=Computers &amp; Graphics|volume=24|issue=3|pages=343–352|doi=10.1016/S0097-8493(00)00030-3|issn=0097-8493}}&lt;/ref&gt;

==== Minimum Intensity Projection (MinIP) ====
Minimum Intensity Projection  highlight the lowest intensity values along a specific projection path. MinIP is particularly beneficial in visualizing structures with low attenuation or density, such as airways in lung imaging. By emphasizing low-intensity features, MinIP can enhance the visibility of subtle details and abnormalities that might be overshadowed in other types of reconstructions. This technique is commonly employed in pulmonary studies to improve the assessment of bronchial structures and airway abnormalities.&lt;ref name=&quot;auto&quot;&gt;{{Cite journal|last=Ghonge|first=NitinP|last2=Chowdhury|first2=Veena|date=2018|title=Minimum-intensity projection images in high-resolution computed tomography lung: Technology update|url=https://journals.lww.com/10.4103/lungindia.lungindia_489_17|journal=Lung India|language=en|volume=35|issue=5|pages=439|doi=10.4103/lungindia.lungindia_489_17|issn=0970-2113|pmc=PMC6120307|pmid=30168468}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Hayabuchi|first=Yasunobu|last2=Inoue|first2=Miki|last3=Watanabe|first3=Noriko|last4=Sakata|first4=Miho|last5=Nabo|first5=Manal Mohamed Helmy|last6=Kagami|first6=Shoji|date=2011-06|title=Minimum-intensity projection of multidetector-row computed tomography for assessment of pulmonary hypertension in children with congenital heart disease|url=https://doi.org/10.1016/j.ijcard.2010.01.008|journal=International Journal of Cardiology|volume=149|issue=2|pages=192–198|doi=10.1016/j.ijcard.2010.01.008|issn=0167-5273}}&lt;/ref&gt;

==== Average Intensity Projection ====
In Average Intensity Projection (AIP), the image is displaying the average attenuation of each voxel within the selected volume.&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-540-89232-8|title=CT of the Acute Abdomen|date=2011|publisher=Springer Berlin Heidelberg|isbn=978-3-540-89231-1|editor-last=Taourel|editor-first=Patrice|series=Medical Radiology|location=Berlin, Heidelberg|language=en|doi=10.1007/978-3-540-89232-8}} Pg. 71&lt;/ref&gt; As the slice thickness increases, the image becomes smoother and more akin to conventional projectional radiography.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=xPwG17yFkzcC&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA213&amp;dq=average+intensity+projection+ct&amp;hl=en|title=CT and MR Angiography: Comprehensive Vascular Assessment|last=Rubin|first=Geoffrey D.|last2=Rofsky|first2=Neil M.|date=2012-10-09|publisher=Lippincott Williams &amp; Wilkins|isbn=978-1-4698-0183-4|language=en}}&lt;/ref&gt; AIP is particularly useful for identifying internal structures of solid organs or the walls of hollow structures, such as intestines. 
{| class=&quot;wikitable&quot;
|+Examples of different algorithms of thickening multiplanar reconstructions&lt;ref&gt;{{Cite journal |last1=Dalrymple |first1=Neal C. |last2=Prasad |first2=Srinivasa R. |last3=Freckleton |first3=Michael W. |last4=Chintapalli |first4=Kedar N. |date=September 2005 |title=Informatics in radiology (infoRAD): introduction to the language of three-dimensional imaging with multidetector CT |journal=Radiographics |volume=25 |issue=5 |pages=1409–1428 |doi=10.1148/rg.255055044 |issn=1527-1323 |pmid=16160120}}&lt;/ref&gt;
!Type of projection
!Schematic illustration
!Examples (10&amp;nbsp;mm slabs)
!Description
!Uses
|-
|Average intensity projection (AIP)
|[[File:Average intensity projection.gif|frameless]]
|[[File:Coronal average intensity projection CT thorax.gif|frameless|118x118px]]
|The average attenuation of each voxel is displayed. The image will get smoother as slice thickness increases. It will look more and more similar to conventional [[w:projectional radiography|projectional radiography]] as slice thickness increases.
|Useful for identifying the internal structures of a solid organ or the walls of hollow structures, such as  intestines.
|-
|[[w:Maximum intensity projection|Maximum intensity projection]] (MIP)
|[[File:Maximum intensity projection.gif|frameless]]
|[[File:Coronal maximum intensity projection CT thorax.gif|frameless|118x118px]]
|The voxel with the highest attenuation is displayed. Therefore, high-attenuating structures such as blood vessels filled with contrast media are enhanced.
|Useful for angiographic studies and identification of pulmonary nodules.
|-
|[[w:Minimum intensity projection|Minimum intensity projection]] (MinIP)
|[[File:Minimum intensity projection.gif|frameless]]
|[[File:Coronal minimum intensity projection CT thorax.gif|frameless|117x117px]]
|The voxel with the lowest attenuation is displayed. Therefore, low-attenuating structures such as air spaces are enhanced.
|Useful for assessing the lung parenchyma.&lt;ref name=&quot;auto&quot;/&gt;&lt;ref&gt;{{Cite book|title=CT of the airways|last=Boiselle|first=Phillip M.|last2=Lynch|first2=David A.|date=2008|publisher=Humana Press|isbn=978-1-59745-139-0|series=Contemporary medical imaging|location=Totowa, NJ}} Page 353&lt;/ref&gt;
|}
====  Volume rendering ====
[[File:Volume rendered CT scan of abdominal and pelvic blood vessels.gif|thumb|191x191px|Volume rendered CT scan of abdominal and pelvic blood vessels.]]
A threshold value of radiodensity is set by the operator (e.g., a level that corresponds to bone). With the help of [[w:edge detection|edge detection]] image processing algorithms a 3D model can be constructed from the initial data and displayed on screen. Various thresholds can be used to get multiple models, each anatomical component such as  muscle, bone and cartilage can be differentiated on the basis of different colours given to them. However, this mode of operation cannot show interior structures.&lt;ref&gt;{{Cite journal |last1=Calhoun |first1=Paul S. |last2=Kuszyk |first2=Brian S. |last3=Heath |first3=David G. |last4=Carley |first4=Jennifer C. |last5=Fishman |first5=Elliot K. |date=1999-05-01 |title=Three-dimensional Volume Rendering of Spiral CT Data: Theory and Method |url=https://pubs.rsna.org/doi/full/10.1148/radiographics.19.3.g99ma14745 |journal=RadioGraphics |volume=19 |issue=3 |pages=745–764 |doi=10.1148/radiographics.19.3.g99ma14745 |issn=0271-5333 |pmid=10336201}}&lt;/ref&gt;

Surface rendering is limited technique as it displays only the surfaces that meet a particular threshold density, and which are towards the viewer. However, In [[w:volume rendering|volume rendering]], transparency, colours and [[w:Phong shading|shading]] are used which makes it easy to present a volume in a single image. For example, Pelvic bones could be displayed as semi-transparent, so that, even viewing at an oblique angle one part of the image does not hide another.&lt;ref&gt;{{Cite journal |last1=van Ooijen |first1=P. M. A. |last2=van Geuns |first2=R. J. M. |last3=Rensing |first3=B. J. W. M. |last4=Bongaerts |first4=A. H. H. |last5=de Feyter |first5=P. J. |last6=Oudkerk |first6=M. |date=January 2003 |title=Noninvasive Coronary Imaging Using Electron Beam CT: Surface Rendering Versus Volume Rendering |url=http://www.ajronline.org/doi/10.2214/ajr.180.1.1800223 |journal=American Journal of Roentgenology |language=en |volume=180 |issue=1 |pages=223–226 |doi=10.2214/ajr.180.1.1800223 |issn=0361-803X |pmid=12490509}}&lt;/ref&gt;

=== Image quality ===
The image quality in computed tomography depends upon the fidelity with which the generated images faithfully represent the attenuation values of X-ray beams as they pass through body tissues, as manifested in the resulting CT image. Image quality encompasses the accurate replication of fine details (Spatial Resolution) and minute discrepancies in attenuation (Contrast Resolution) within the depicted image.

==== Dose versus image quality ====
An important issue within radiology today is how to reduce the radiation dose during CT examinations without compromising the image quality. In general, higher radiation doses result in higher-resolution images,&lt;ref name=&quot;Crowther&quot;&gt;{{Cite journal |last1=R. A. Crowther |last2=D. J. DeRosier |last3=A. Klug |year=1970 |title=The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy |journal=Proc. R. Soc. Lond. A |volume=317 |issue=1530 |pages=319–340 |bibcode=1970RSPSA.317..319C |doi=10.1098/rspa.1970.0119 |s2cid=122980366}}&lt;/ref&gt; while lower doses lead to increased image noise and unsharp images. However, increased dosage raises the adverse side effects, including the risk of [[w:radiation-induced cancer|radiation-induced cancer]] – a four-phase abdominal CT gives the same radiation dose as 300 chest X-rays.&lt;ref&gt;{{Cite journal |last1=Nickoloff |first1=Edward L. |last2=Alderson |first2=Philip O. |date=August 2001 |title=Radiation Exposures to Patients from CT: Reality, Public Perception, and Policy |url=http://www.ajronline.org/doi/10.2214/ajr.177.2.1770285 |journal=American Journal of Roentgenology |language=en |volume=177 |issue=2 |pages=285–287 |doi=10.2214/ajr.177.2.1770285 |issn=0361-803X |pmid=11461846}}&lt;/ref&gt; Several methods that can reduce the exposure to ionizing radiation during a CT scan exist.

# New software technology can significantly reduce the required radiation dose. New [[w:Iterative reconstruction|iterative]] [[w:tomographic reconstruction|tomographic reconstruction]] algorithms (''e.g.'', [[w:SAMV (algorithm)|iterative Sparse Asymptotic Minimum Variance]]) could offer [[w:Super-resolution imaging|super-resolution]] without requiring higher radiation dose.&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=hclVAAAAMAAJ&amp;q=iterative+construction+gives+super+resolution |title=Proceedings |date=1995 |publisher=IEEE |page=10 |isbn=9780780324985 |language=en}}&lt;/ref&gt;
# Individualize the examination and adjust the radiation dose to the body type and body organ examined. Different body types and organs require different amounts of radiation.&lt;ref&gt;{{Cite web |title=Radiation – Effects on organs of the body (somatic effects) |url=https://www.britannica.com/science/radiation |access-date=2021-03-21 |website=Encyclopedia Britannica |language=en}}&lt;/ref&gt;
# Higher resolution is not always suitable, such as detection of small pulmonary masses.&lt;ref&gt;{{Cite journal |last=Simpson G |year=2009 |title=Thoracic computed tomography: principles and practice |journal=Australian Prescriber |volume=32 |issue=4 |page=4 |doi=10.18773/austprescr.2009.049 |doi-access=free}}&lt;/ref&gt;

==== Artifacts ====
Although images produced by CT are generally faithful representations of the scanned volume, the technique is susceptible to a number of [[w:artifact (error)#Medical imaging|artifacts]], such as the following:&lt;ref name=&quot;ref1&quot; /&gt;&lt;ref&gt;{{Cite journal |last1=Bhowmik |first1=Ujjal Kumar |last2=Zafar Iqbal, M. |last3=Adhami, Reza R. |date=28 May 2012 |title=Mitigating motion artifacts in FDK based 3D Cone-beam Brain Imaging System using markers |journal=Central European Journal of Engineering |volume=2 |issue=3 |pages=369–382 |bibcode=2012CEJE....2..369B |doi=10.2478/s13531-012-0011-7 |doi-access=free}}&lt;/ref&gt;
[[File:Aufhaertungsartefakte in der CT durch HTEP 86W - CT axial - 001.jpg|thumb|Metal artifact seen on the right side to a hip prosthesis.]]
Streaks are often seen around materials that block most X-rays, such as metal or bone. Numerous factors contribute to these streaks: under sampling, photon starvation, motion, beam hardening, and [[w:Compton scatter|Compton scatter]]. This type of artifact commonly occurs in the posterior fossa of the brain, or if there are metal implants. The streaks can be reduced using newer reconstruction techniques.&lt;ref name=&quot;P. Jin and C. A. Bouman and K. D. Sauer 2013&quot;&gt;{{Cite journal |last1=P. Jin |last2=C. A. Bouman |last3=K. D. Sauer |year=2013 |title=A Method for Simultaneous Image Reconstruction and Beam Hardening Correction |url=https://engineering.purdue.edu/~bouman/publications/pdf/mic2013.pdf |url-status=dead |journal=IEEE Nuclear Science Symp. &amp; Medical Imaging Conf., Seoul, Korea, 2013 |archive-url=https://web.archive.org/web/20140606234132/https://engineering.purdue.edu/~bouman/publications/pdf/mic2013.pdf |archive-date=2014-06-06 |access-date=2014-04-23}}&lt;/ref&gt; Approaches such as metal artifact reduction (MAR) can also reduce this artifact.&lt;ref&gt;{{Cite journal |vauthors=Boas FE, Fleischmann D |year=2011 |title=Evaluation of Two Iterative Techniques for Reducing Metal Artifacts in Computed Tomography |journal=Radiology |volume=259 |issue=3 |pages=894–902 |doi=10.1148/radiol.11101782 |pmid=21357521}}&lt;/ref&gt;&lt;ref name=&quot;mouton13survey&quot;&gt;{{Cite journal |last1=Mouton, A. |last2=Megherbi, N. |last3=Van Slambrouck, K. |last4=Nuyts, J. |last5=Breckon, T.P. |year=2013 |title=An Experimental Survey of Metal Artefact Reduction in Computed Tomography |url=http://www.durham.ac.uk/toby.breckon/publications/papers/mouton13survey.pdf |journal=Journal of X-Ray Science and Technology |volume=21 |issue=2 |pages=193–226 |doi=10.3233/XST-130372 |pmid=23694911 |hdl=1826/8204}}&lt;/ref&gt; MAR techniques include spectral imaging, where CT images are taken with [[w:photons|photons]] of different energy levels, and then synthesized into [[w:monochromatic|monochromatic]] images with special software such as GSI (Gemstone Spectral Imaging).&lt;ref name=&quot;PessisCampagna2013&quot;&gt;{{Cite journal |last1=Pessis |first1=Eric |last2=Campagna |first2=Raphaël |last3=Sverzut |first3=Jean-Michel |last4=Bach |first4=Fabienne |last5=Rodallec |first5=Mathieu |last6=Guerini |first6=Henri |last7=Feydy |first7=Antoine |last8=Drapé |first8=Jean-Luc |year=2013 |title=Virtual Monochromatic Spectral Imaging with Fast Kilovoltage Switching: Reduction of Metal Artifacts at CT |journal=RadioGraphics |volume=33 |issue=2 |pages=573–583 |doi=10.1148/rg.332125124 |issn=0271-5333 |pmid=23479714 |doi-access=free}}&lt;/ref&gt;

Partial volume effect appears as &quot;blurring&quot; of edges. It is due to the scanner being unable to differentiate between a small amount of high-density material (e.g., bone) and a larger amount of lower density  (e.g., cartilage).&lt;ref&gt;{{Cite journal |last1=González Ballester |first1=Miguel Angel |last2=Zisserman |first2=Andrew P. |last3=Brady |first3=Michael |date=December 2002 |title=Estimation of the partial volume effect in MRI |journal=Medical Image Analysis |volume=6 |issue=4 |pages=389–405 |doi=10.1016/s1361-8415(02)00061-0 |issn=1361-8415 |pmid=12494949}}&lt;/ref&gt; The reconstruction assumes that the X-ray attenuation within each voxel is homogeneous; this may not be the case at sharp edges. This is most commonly seen in the z-direction (craniocaudal direction), due to the conventional use of highly [[w:isotropic|anisotropic]] voxels, which have a much lower out-of-plane resolution, than in-plane resolution. This can be partially overcome by scanning using thinner slices, or an isotropic acquisition on a modern scanner.&lt;ref&gt;{{Cite journal |last1=Goldszal |first1=Alberto F. |last2=Pham |first2=Dzung L. |date=2000-01-01 |title=Volumetric Segmentation |journal=Handbook of Medical Imaging |language=en |pages=185–194 |doi=10.1016/B978-012077790-7/50016-3 |isbn=978-0-12-077790-7}}&lt;/ref&gt;

Ring artifact probably the most common mechanical artifact, the image of one or many &quot;rings&quot; appears within an image. They are usually caused by the variations in the response from individual elements in a two dimensional X-ray detector due to defect or miscalibration.&lt;ref name=&quot;Jha&quot;&gt;{{Cite journal |last=Jha |first=Diwaker |date=2014 |title=Adaptive center determination for effective suppression of ring artifacts in tomography images |journal=Applied Physics Letters |volume=105 |issue=14 |pages=143107 |bibcode=2014ApPhL.105n3107J |doi=10.1063/1.4897441}}&lt;/ref&gt; This phenomenon is more frequently encountered in third-generation CT scanners equipped with solid-state detectors. These artifacts manifest as complete circles in sequential scans, or partial rings in helical CT scans.&lt;ref&gt;{{Cite journal|last=Ramasamy|first=Akilesh|last2=Madhan|first2=Balasubramanian|last3=Krishnan|first3=Balasubramanian|date=2018-08-03|title=Ring artefacts in cranial CT|url=https://casereports.bmj.com/lookup/doi/10.1136/bcr-2018-226097|journal=BMJ Case Reports|language=en|pages=bcr–2018–226097|doi=10.1136/bcr-2018-226097|issn=1757-790X|pmc=PMC6078223|pmid=30076165}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|title=The essential physics of medical imaging|date=2012|publisher=Wolters Kluwer, Lippincott Williams &amp; Wilkins|isbn=978-0-7817-8057-5|editor-last=Bushberg|editor-first=Jerrold T.|edition=3. ed|location=Philadelphia}} p. 373&lt;/ref&gt; Ring artifacts can largely be reduced by intensity normalization, also referred to as flat field correction.&lt;ref name=&quot;vvn15&quot;&gt;{{Cite journal |last1=Van Nieuwenhove |first1=V |last2=De Beenhouwer |first2=J |last3=De Carlo |first3=F |last4=Mancini |first4=L |last5=Marone |first5=F |last6=Sijbers |first6=J |date=2015 |title=Dynamic intensity normalization using eigen flat fields in X-ray imaging |url=http://www.zora.uzh.ch/id/eprint/120683/1/oe-23-21-27975.pdf |journal=Optics Express |volume=23 |issue=21 |pages=27975–27989 |bibcode=2015OExpr..2327975V |doi=10.1364/oe.23.027975 |pmid=26480456 |doi-access=free |hdl=10067/1302930151162165141}}&lt;/ref&gt; Remaining rings can be suppressed by a transformation to polar space, where they become linear stripes.&lt;ref name=&quot;Jha&quot; /&gt; A comparative evaluation of ring artefact reduction on X-ray tomography images showed that the method of Sijbers and Postnov can effectively suppress ring artefacts.&lt;ref name=&quot;jsap&quot;&gt;{{Cite journal |vauthors=Sijbers J, Postnov A |date=2004 |title=Reduction of ring artefacts in high resolution micro-CT reconstructions |journal=Phys Med Biol |volume=49 |issue=14 |pages=N247–53 |doi=10.1088/0031-9155/49/14/N06 |pmid=15357205 |s2cid=12744174}}&lt;/ref&gt;[[File:Bewegungsartefakte im CCT 72W - CT - 001.jpg|thumb|191x191px|Motion Artifact]]Noise appears as grain on the image and is caused by a low signal to noise ratio. This occurs more commonly when a thin slice thickness is used. It can also occur when the power supplied to the X-ray tube is insufficient to penetrate the anatomy.&lt;ref&gt;{{Cite book |last1=Newton |first1=Thomas H. |url=https://books.google.com/books?id=2mxsAAAAMAAJ&amp;q=noise+in+computed+tomography |title=Radiology of the Skull and Brain: Technical aspects of computed tomography |last2=Potts |first2=D. Gordon |date=1971 |publisher=Mosby |isbn=978-0-8016-3662-2 |pages=3941–3950 |language=en}}&lt;/ref&gt;

Windmill artifacts is seen as a streaking appearances which can occur when the detectors intersect the reconstruction plane. This can be reduced with filters or a reduction in pitch.&lt;ref&gt;{{Cite book |last1=Brüning |first1=R. |url=https://books.google.com/books?id=ImOlZNOk25sC&amp;q=windmill+artifact+ct&amp;pg=PA44 |title=Protocols for Multislice CT |last2=Küttner |first2=A. |last3=Flohr |first3=T. |date=2006-01-16 |publisher=Springer Science &amp; Business Media |isbn=978-3-540-27273-1 |language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last=Peh |first=Wilfred C. G. |url=https://books.google.com/books?id=sZswDwAAQBAJ&amp;q=windmill+artifact+ct&amp;pg=PA49 |title=Pitfalls in Musculoskeletal Radiology |date=2017-08-11 |publisher=Springer |isbn=978-3-319-53496-1 |language=en}}&lt;/ref&gt;

Beam hardening artefact can give a &quot;cupped appearance&quot; when grayscale is visualized as height. It occurs because conventional sources, like X-ray tubes emit a polychromatic spectrum. Photons of higher [[w:photon energy|photon energy]] levels are typically attenuated less. Because of this, the mean energy of the spectrum increases when passing the object, often described as getting &quot;harder&quot;. This leads to an effect increasingly underestimating material thickness, if not corrected. Many algorithms exist to correct for this artifact. They can be divided in mono- and multi-material methods.&lt;ref name=&quot;P. Jin and C. A. Bouman and K. D. Sauer 2013&quot; /&gt;&lt;ref&gt;{{Cite journal |vauthors=Van de Casteele E, Van Dyck D, Sijbers J, Raman E |year=2004 |title=A model-based correction method for beam hardening artefacts in X-ray microtomography |journal=Journal of X-ray Science and Technology |volume=12 |issue=1 |pages=43–57 |citeseerx=10.1.1.460.6487}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal |vauthors=Van Gompel G, Van Slambrouck K, Defrise M, Batenburg KJ, Sijbers J, Nuyts J |year=2011 |title=Iterative correction of beam hardening artifacts in CT |journal=Medical Physics |volume=38 |issue=1 |pages=36–49 |bibcode=2011MedPh..38S..36V |citeseerx=10.1.1.464.3547 |doi=10.1118/1.3577758 |pmid=21978116}}&lt;/ref&gt;

Motion artifact refers to unwanted distortions of the images caused by patient motion which can be voluntary on involuntary during the scanning process.

== Advantages ==
CT scan has several advantages over traditional [[w:two-dimensional space|two-dimensional]] medical [[w:radiography|radiography]]. First, CT eliminates the superimposition of images of structures outside the area of interest.&lt;ref&gt;{{Cite book |last1=Mikla |first1=Victor I. |url=https://books.google.com/books?id=Y81JrnVA_5sC&amp;q=ct+scan+removes+superimposition&amp;pg=PA37 |title=Medical Imaging Technology |last2=Mikla |first2=Victor V. |date=2013-08-23 |publisher=Elsevier |isbn=978-0-12-417036-0 |page=37 |language=en}}&lt;/ref&gt; Second, CT scans have greater [[w:image resolution|image resolution]], enabling examination of finer details. CT can distinguish between [[w:tissue (biology)|tissues]] that differ in radiographic [[w:density|density]] by  1% or less.&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=rOppAAAAMAAJ&amp;q=CT+can+distinguish+between+tissue |title=Radiology for the Dental Professional |publisher=Elsevier Mosby |year=2008 |isbn=978-0-323-03071-7 |pages=337}}&lt;/ref&gt; Third, CT scanning enables multiplanar reformatted imaging: scan data can be visualized in the [[w:transverse plane|transverse (or axial)]], [[w:Coronal plane|coronal]], or [[w:Sagittal plane|sagittal]] plane, depending on the diagnostic task.&lt;ref&gt;{{Cite book |last=Pasipoularides |first=Ares |url=https://books.google.com/books?id=eMKqdIvxEmQC&amp;q=ct+scan+enables+multiple+plane+reformatting&amp;pg=PA595 |title=Heart's Vortex: Intracardiac Blood Flow Phenomena |date=November 2009 |publisher=PMPH-USA |isbn=978-1-60795-033-2 |pages=595 |language=en}}&lt;/ref&gt;

The improved resolution of CT has permitted the development of new investigations. For example, CT [[w:angiography|angiography]] avoids the invasive insertion of a [[w:catheter|catheter]]. CT scanning can perform a [[w:virtual colonoscopy|virtual colonoscopy]] with greater accuracy and less discomfort for the patient than a traditional [[w:colonoscopy|colonoscopy]].&lt;ref name=&quot;Heiken&quot;&gt;{{Cite journal |last1=Heiken |first1=JP |last2=Peterson CM |last3=Menias CO |date=November 2005 |title=Virtual colonoscopy for colorectal cancer screening: current status: Wednesday 5 October 2005, 14:00–16:00 |journal=Cancer Imaging |publisher=International Cancer Imaging Society |volume=5 |issue=Spec No A |pages=S133–S139 |doi=10.1102/1470-7330.2005.0108 |pmc=1665314 |pmid=16361129}}&lt;/ref&gt;&lt;ref name=&quot;pmid16106357&quot;&gt;{{Cite journal |last1=Bielen DJ |last2=Bosmans HT |last3=De Wever LL |last4=Maes |first4=Frederik |last5=Tejpar |first5=Sabine |last6=Vanbeckevoort |first6=Dirk |last7=Marchal |first7=Guy J.F. |display-authors=3 |name-list-style=vanc |date=September 2005 |title=Clinical validation of high-resolution fast spin-echo MR colonography after colon distention with air |journal=J Magn Reson Imaging |volume=22 |issue=3 |pages=400–5 |doi=10.1002/jmri.20397 |pmid=16106357 |doi-access=free |s2cid=22167728}}&lt;/ref&gt; Virtual colonography is far more accurate than a [[w:barium enema|barium enema]] for detection of tumors and uses a lower radiation dose.&lt;ref&gt;{{Cite web |title=CT Colonography |url=https://www.radiologyinfo.org/en/info.cfm?pg=ct_colo |website=Radiologyinfo.org}}&lt;/ref&gt;

CT is a moderate-to-high [[w:radiation|radiation]] diagnostic technique. The radiation dose for a particular examination depends on multiple factors: volume scanned, patient build, number and type of scan protocol, and desired resolution and image quality.&lt;ref&gt;{{Cite journal |vauthors=Žabić S, Wang Q, Morton T, Brown KM |date=March 2013 |title=A low dose simulation tool for CT systems with energy integrating detectors |journal=Medical Physics |volume=40 |issue=3 |pages=031102 |bibcode=2013MedPh..40c1102Z |doi=10.1118/1.4789628 |pmid=23464282}}&lt;/ref&gt; Two helical CT scanning parameters, tube current and pitch, can be adjusted easily and have a profound effect on radiation. CT scanning is more accurate than two-dimensional radiographs in evaluating anterior interbody fusion,  although they may still over-read the extent of fusion.&lt;ref&gt;Brian R. Subach M.D., F.A.C.S et al.[http://www.spinemd.com/publications/articles/reliability-and-accuracy-of-fine-cut-computed-tomography-scans-to-determine-the-status-of-anterior-interbody-usions-with-metallic-cages &quot;Reliability and accuracy of fine-cut computed tomography scans to determine the status of anterior interbody fusions with metallic cages&quot;] {{webarchive|url=https://web.archive.org/web/20121208184918/http://www.spinemd.com/publications/articles/reliability-and-accuracy-of-fine-cut-computed-tomography-scans-to-determine-the-status-of-anterior-interbody-usions-with-metallic-cages |date=2012-12-08 }}&lt;/ref&gt;

== Adverse effects ==

=== Contrast reactions ===
The administration of contrast agents carries potential risks, as adverse reactions may occur. While most serious reactions are typically observed after intravascular injection, adverse effects may still manifest after oral or intra-cavitary administration, as some contrast medium molecules may be absorbed into the circulation. Contrast reactions reported in CT scans are classified into three severity levels: Mild, Moderate, and Severe Reactions. 
{| class=&quot;sortable wikitable&quot; style=&quot;float: right; margin-left:15px; text-align:center&quot;
|+Types of Contrast Reactions
!Type
!Severity
!Symptoms may include
!Medical attention
|-
|Mild Reaction
|Low
|nausea, headache, vomiting, flushing, pruritus, and a metallic taste.
|Generally not required, typically resolve on their own.
|-
|Moderate Reactions
|Intermediate
|urticaria, severe vomiting, facial edema, bronchospasm,  laryngeal edema, and vasovagal attacks.
|Treatment approach varies based on the specific symptoms.
|-
|Severe Reaction
|High
|respiratory arrest, cardiac arrest, pulmonary edema, convulsions, and cardiogenic shock.
|Immediate medical attention is essential
|}
In the United States half of CT scans are [[w:contrast CT|contrast CT]]s using intravenously injected [[w:radiocontrast agent|radiocontrast agent]]s.&lt;ref name=&quot;Nam2006&quot; /&gt; The most common reactions from these agents are mild, including nausea, vomiting, and an itching rash. Severe life-threatening reactions may rarely occur.&lt;ref name=&quot;Contrast2005&quot;&gt;{{Cite journal |last=Christiansen C |date=2005-04-15 |title=X-ray contrast media – an overview |journal=Toxicology |volume=209 |issue=2 |pages=185–7 |doi=10.1016/j.tox.2004.12.020 |pmid=15767033}}&lt;/ref&gt; Overall reactions occur in 1 to 3% with [[w:nonionic contrast|nonionic contrast]] and 4 to 12% of people with [[w:ionic contrast|ionic contrast]].&lt;ref name=&quot;Wang2011&quot; /&gt; Skin rashes may appear within a week to 3% of people.&lt;ref name=&quot;Contrast2005&quot; /&gt;

The old [[w:radiocontrast agent|radiocontrast agent]]s caused [[w:anaphylaxis|anaphylaxis]] in 1% of cases while the newer, low-osmolar agents cause reactions in 0.01–0.04% of cases.&lt;ref name=&quot;Contrast2005&quot; /&gt;&lt;ref name=&quot;Drug01&quot;&gt;{{Cite journal |vauthors=Drain KL, Volcheck GW |year=2001 |title=Preventing and managing drug-induced anaphylaxis |journal=Drug Safety |volume=24 |issue=11 |pages=843–53 |doi=10.2165/00002018-200124110-00005 |pmid=11665871 |s2cid=24840296}}&lt;/ref&gt; Death occurs in about 2 to 30 people per 1,000,000 administrations, with newer agents being safer.&lt;ref name=&quot;Wang2011&quot;&gt;{{Cite journal |vauthors=Wang H, Wang HS, Liu ZP |date=October 2011 |title=Agents that induce pseudo-allergic reaction |journal=Drug Discov Ther |volume=5 |issue=5 |pages=211–9 |doi=10.5582/ddt.2011.v5.5.211 |pmid=22466368 |s2cid=19001357}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=bEvnfm7V-LIC&amp;pg=PA187 |title=Anaphylaxis and hypersensitivity reactions |date=2010-12-09 |publisher=Humana Press |isbn=978-1-60327-950-5 |editor-last=Castells |editor-first=Mariana C. |location=New York |page=187}}&lt;/ref&gt; There is a higher risk of mortality in those who are female, elderly or in poor health, usually secondary to either anaphylaxis or [[w:acute kidney injury|acute kidney injury]].&lt;ref name=&quot;Nam2006&quot;&gt;{{Cite journal |vauthors=Namasivayam S, Kalra MK, Torres WE, Small WC |date=Jul 2006 |title=Adverse reactions to intravenous iodinated contrast media: a primer for radiologists |journal=Emergency Radiology |volume=12 |issue=5 |pages=210–5 |doi=10.1007/s10140-006-0488-6 |pmid=16688432 |s2cid=28223134}}&lt;/ref&gt;

The contrast agent may induce [[w:contrast-induced nephropathy|contrast-induced nephropathy]].&lt;ref name=&quot;Contrast2009&quot;&gt;{{Cite journal |vauthors=Hasebroock KM, Serkova NJ |date=April 2009 |title=Toxicity of MRI and CT contrast agents |journal=Expert Opinion on Drug Metabolism &amp; Toxicology |volume=5 |issue=4 |pages=403–16 |doi=10.1517/17425250902873796 |pmid=19368492 |s2cid=72557671}}&lt;/ref&gt; This occurs in 2 to 7% of people who receive these agents, with greater risk in those who have pre-existing [[w:kidney failure|kidney failure]],&lt;ref name=&quot;Contrast2009&quot; /&gt; pre-existing [[w:diabetes mellitus|diabetes]], or reduced intravascular volume. People with mild kidney impairment are usually advised to ensure full hydration for several hours before and after the injection. For moderate kidney failure, the use of  [[w:iodinated contrast|iodinated contrast]] should be avoided; this may mean using an alternative technique instead of CT. Those with severe [[w:kidney failure|kidney failure]] requiring [[w:Kidney dialysis|dialysis]] require less strict precautions, as their kidneys have so little function remaining that any further damage would not be noticeable and the dialysis will remove the contrast agent; it is normally recommended, however, to arrange dialysis as soon as possible following contrast administration to minimize any adverse effects of the contrast.

In addition to the use of intravenous contrast, orally administered contrast agents are frequently used when examining the abdomen.&lt;ref&gt;{{Cite journal |last1=Rawson |first1=James V. |last2=Pelletier |first2=Allen L. |date=2013-09-01 |title=When to Order Contrast-Enhanced CT |url=https://www.aafp.org/afp/2013/0901/p312.html |journal=American Family Physician |volume=88 |issue=5 |pages=312–316 |issn=0002-838X |pmid=24010394}}&lt;/ref&gt; These are frequently the same as the intravenous contrast agents, merely diluted to approximately 10% of the concentration. However, oral alternatives to iodinated contrast exist, such as very dilute (0.5–1% w/v) [[w:barium sulfate|barium sulfate]] suspensions. Dilute barium sulfate has the advantage that it does not cause allergic-type reactions or kidney failure, but cannot be used in patients with suspected bowel perforation or suspected bowel injury, as leakage of barium sulfate from damaged bowel can cause fatal [[w:peritonitis|peritonitis]].&lt;ref&gt;{{Cite book |last1=Thomsen |first1=Henrik S. |url=https://books.google.com/books?id=Bun1CAAAQBAJ&amp;q=intravenous+contrast+in+ct |title=Trends in Contrast Media |last2=Muller |first2=Robert N. |last3=Mattrey |first3=Robert F. |date=2012-12-06 |publisher=Springer Science &amp; Business Media |isbn=978-3-642-59814-2 |language=en}}&lt;/ref&gt;

Side effects from [[w:contrast agent|contrast agent]]s, administered [[w:Intravenous therapy|intravenously]]  in some CT scans, might impair [[w:kidney|kidney]] performance in patients with [[w:kidney disease|kidney disease]], although this risk is now believed to be lower than previously thought.&lt;ref&gt;{{Cite journal |last=Davenport |first=Matthew |year=2020 |title=Use of Intravenous Iodinated Contrast Media in Patients with Kidney Disease: Consensus Statements from the American College of Radiology and the National Kidney Foundation |journal=Radiology |volume=294 |issue=3 |pages=660–668 |doi=10.1148/radiol.2019192094 |pmid=31961246 |doi-access=free}}&lt;/ref&gt;&lt;ref name=&quot;Contrast2009&quot; /&gt;

==== Extravasation ====
Extravasation refers to the unintended leakage of contrast medium, from the intravascular space into the surrounding tissues.&lt;ref&gt;{{Cite journal|last=Alexander|first=Leon|last2=Sheikh Khalifa Medical City, Division of Plastic Surgery, Department of Surgery, Abu Dhabi, UAE|date=2020-09-01|title=Extravasation Injuries: A Trivial Injury Often Overlooked with Disastrous Consequences|url=http://wjps.ir/article-1-637-en.html|journal=WORLD JOURNAL OF PLASTIC SURGERY|language=en|volume=9|issue=3|pages=326–330|doi=10.29252/wjps.9.3.326|issn=2228-7914|pmc=PMC7734938|pmid=33330011}}&lt;/ref&gt; This phenomenon can occur due to various mechanisms during medical procedures involving injection of contrast medium. The fluid can seep into perivascular tissue either by extraluminal dislocation of cannula. Additionally, a leak may manifest at the puncture site of a properly positioned cannula, causing extravasation. The direct impact of pressure from the jet against the vessel wall can lead to vessel disruption and, consequently, extravasation.&lt;ref&gt;{{Cite journal|last=Sakellariou|first=Sophia|last2=Li|first2=Wenguang|last3=Paul|first3=Manosh C|last4=Roditi|first4=Giles|date=2016-12|title=Rôle of contrast media viscosity in altering vessel wall shear stress and relation to the risk of contrast extravasations|url=https://linkinghub.elsevier.com/retrieve/pii/S1350453316302168|journal=Medical Engineering &amp; Physics|language=en|volume=38|issue=12|pages=1426–1433|doi=10.1016/j.medengphy.2016.09.016}}&lt;/ref&gt; Extravasation can arise from both manual hand injection and injections administered using a press-injector. Patients at a greater risk of experiencing extravasation encompass the elderly, infants, children, individuals with impaired consciousness, and those with pre-existing vascular conditions.&lt;ref&gt;{{Cite journal|last=Wang|first=Carolyn L.|last2=Cohan|first2=Richard H.|last3=Ellis|first3=James H.|last4=Adusumilli|first4=Saroja|last5=Dunnick|first5=N. Reed|date=2007-04|title=Frequency, Management, and Outcome of Extravasation of Nonionic Iodinated Contrast Medium in 69 657 Intravenous Injections|url=http://pubs.rsna.org/doi/10.1148/radiol.2431060554|journal=Radiology|language=en|volume=243|issue=1|pages=80–87|doi=10.1148/radiol.2431060554|issn=0033-8419}}&lt;/ref&gt;

Moderate extravasation is characterized by symptoms, such as skin blistering, progressive edema, or ulceration. Close monitoring is advised, and physician assessment is recommended to evaluate for potential neurovascular compromise. This assessment includes checking peripheral pulse and assessing sensation distal to the affected limb. Severe extravasation represents a critical condition with the potential for neurovascular compromise, signs of tissue necrosis, or compartment syndrome. Urgent surgical attention, specifically emergency fasciotomy, is necessary to alleviate pressure within the affected compartment and prevent further complications. &lt;ref&gt;{{Cite journal|last=Liu|first=Wanli|last2=Wang|first2=Pinghu|last3=Zhu|first3=Hui|last4=Tang|first4=Hui|last5=Guan|first5=Hongmei|last6=Wang|first6=Xiaoying|last7=Wang|first7=Chengxiang|last8=Qiu|first8=Yao|last9=He|first9=Lianxiang|date=2023-10-25|title=Contrast media extravasation injury: a prospective observational cohort study|url=https://doi.org/10.1186/s40001-023-01444-5|journal=European Journal of Medical Research|volume=28|issue=1|pages=458|doi=10.1186/s40001-023-01444-5|issn=2047-783X|pmc=PMC10598951|pmid=37880738}}&lt;/ref&gt;

=== Scan dose ===
{| class=&quot;sortable wikitable&quot; style=&quot;float: right; margin-left:15px; text-align:center&quot;
|-
!Examination
!Typical [[w:Effective dose (radiation safety)|effective &lt;br /&gt; dose]] ([[w:Sievert|mSv]])&lt;br /&gt; to the whole body
!Typical [[w:Absorbed dose|absorbed &lt;br /&gt; dose]] ([[w:Gray (unit)|mGy]])&lt;br /&gt; to the organ in question
|-
|Annual background radiation
|2.4&lt;ref name=&quot;background&quot; /&gt;
|2.4&lt;ref name=&quot;background&quot; /&gt;
|-
|Chest X-ray
|0.02&lt;ref name=&quot;FDADose&quot;&gt;{{Cite web |year=2009 |title=What are the Radiation Risks from CT? |url=https://www.fda.gov/radiation-emittingproducts/radiationemittingproductsandprocedures/medicalimaging/medicalX-rays/ucm115329.htm |url-status=live |archive-url=https://web.archive.org/web/20131105050317/https://www.fda.gov/Radiation-EmittingProducts/RadiationEmittingProductsandProcedures/MedicalImaging/MedicalX-Rays/ucm115329.htm |archive-date=2013-11-05 |website=Food and Drug Administration}}&lt;/ref&gt;
|0.01–0.15&lt;ref name=&quot;crfdr&quot; /&gt;
|-
|Head CT
|1–2&lt;ref name=&quot;Furlow2010&quot; /&gt;
|56&lt;ref name=&quot;nrpb2005&quot;&gt;Shrimpton, P.C; Miller, H.C; Lewis, M.A; Dunn, M.  [http://www.hpa.org.uk/web/HPAwebFile/HPAweb_C/1194947420292 Doses from Computed Tomography (CT) examinations in the UK – 2003 Review] {{webarchive|url=https://web.archive.org/web/20110922122151/http://www.hpa.org.uk/web/HPAwebFile/HPAweb_C/1194947420292 |date=2011-09-22 }}&lt;/ref&gt;
|-
|Screening [[mammography]]
|0.4&lt;ref name=&quot;Risk2011&quot;&gt;{{Cite journal |last1=Davies |first1=H. E. |last2=Wathen, C. G. |last3=Gleeson, F. V. |date=25 February 2011 |title=The risks of radiation exposure related to diagnostic imaging and how to minimise them |journal=BMJ |volume=342 |issue=feb25 1 |pages=d947 |doi=10.1136/bmj.d947 |pmid=21355025 |s2cid=206894472}}&lt;/ref&gt;
|3&lt;ref name=&quot;Brenner2007&quot;&gt;{{Cite journal|date=November 2007|title=Computed tomography – an increasing source of radiation exposure|url=http://www.columbia.edu/~djb3/papers/nejm1.pdf|journal=N. Engl. J. Med.|volume=357|issue=22|pages=2277–84|doi=10.1056/NEJMra072149|pmid=18046031|archive-url=https://web.archive.org/web/20160304060542/http://www.columbia.edu/~djb3/papers/nejm1.pdf|archive-date=2016-03-04|vauthors=Brenner DJ, Hall EJ|url-status=live|s2cid=2760372}}&lt;/ref&gt;&lt;ref name=&quot;crfdr&quot; /&gt;
|-
|Abdominal CT
|8&lt;ref name=&quot;FDADose&quot; /&gt;
|14&lt;ref name=&quot;nrpb2005&quot; /&gt;
|-
|Chest CT
|5–7&lt;ref name=&quot;Furlow2010&quot; /&gt;
|13&lt;ref name=&quot;nrpb2005&quot; /&gt;
|-
|[[Virtual colonoscopy|CT colonography]]
|6–11&lt;ref name=&quot;Furlow2010&quot; /&gt;
|
|-
|Chest, abdomen and pelvis CT
|9.9&lt;ref name=&quot;nrpb2005&quot; /&gt;
|12&lt;ref name=&quot;nrpb2005&quot; /&gt;
|-
|Cardiac CT angiogram
|9–12&lt;ref name=&quot;Furlow2010&quot; /&gt;
|40–100&lt;ref name=&quot;crfdr&quot; /&gt;
|-
|[[Barium enema]]
|15&lt;ref name=&quot;Brenner2007&quot; /&gt;
|15&lt;ref name=&quot;crfdr&quot; /&gt;
|-
|Neonatal abdominal CT
|20&lt;ref name=&quot;Brenner2007&quot; /&gt;
|20&lt;ref name=&quot;crfdr&quot; /&gt;
|-
| colspan=&quot;3&quot; |
|}
The magnitude of radiation exposure encountered by patients undergoing computed tomography examinations depends on the scanner's design,&lt;ref&gt;{{Cite journal|last=Jaffe|first=Tracy A.|last2=Yoshizumi|first2=Terry T.|last3=Toncheva|first3=Greta|last4=Anderson-Evans|first4=Colin|last5=Lowry|first5=Carolyn|last6=Miller|first6=Chad M.|last7=Nelson|first7=Rendon C.|last8=Ravin|first8=Carl E.|date=2009-10|title=Radiation Dose for Body CT Protocols: Variability of Scanners at One Institution|url=https://www.ajronline.org/doi/10.2214/AJR.09.2330|journal=American Journal of Roentgenology|language=en|volume=193|issue=4|pages=1141–1147|doi=10.2214/AJR.09.2330|issn=0361-803X}}&lt;/ref&gt; and the factors chosen by the radiology technologist including current, voltage, scan field, scan duration, filtration, rotation angle, collimation, and section thickness and spacing, collectively contribute to the overall cumulative dose.&lt;ref&gt;{{Cite journal|last=Rothenberg|first=L N|last2=Pentlow|first2=K S|date=1992-11|title=Radiation dose in CT.|url=http://pubs.rsna.org/doi/10.1148/radiographics.12.6.1439023|journal=RadioGraphics|language=en|volume=12|issue=6|pages=1225–1243|doi=10.1148/radiographics.12.6.1439023|issn=0271-5333}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Kalra|first=Mannudeep K.|last2=Maher|first2=Michael M.|last3=Toth|first3=Thomas L.|last4=Hamberg|first4=Leena M.|last5=Blake|first5=Michael A.|last6=Shepard|first6=Jo-Anne|last7=Saini|first7=Sanjay|date=2004-03|title=Strategies for CT Radiation Dose Optimization|url=http://pubs.rsna.org/doi/10.1148/radiol.2303021726|journal=Radiology|language=en|volume=230|issue=3|pages=619–628|doi=10.1148/radiol.2303021726|issn=0033-8419}}&lt;/ref&gt; These considerations are important for optimizing both equipment configurations and scanning protocols to have a balance between achieving diagnostic precision and minimizing radiation exposure to patients.&lt;ref&gt;{{Cite journal|last=Toth|first=Thomas L.|date=2002-04-01|title=Dose reduction opportunities for CT scanners|url=https://doi.org/10.1007/s00247-002-0678-7|journal=Pediatric Radiology|language=en|volume=32|issue=4|pages=261–267|doi=10.1007/s00247-002-0678-7|issn=1432-1998}}&lt;/ref&gt;

The table reports average radiation exposures; however, there can be a wide variation in radiation doses between similar scan types, where the highest dose could be as much as 22 times higher than the lowest dose.&lt;ref name=&quot;Furlow2010&quot; /&gt;  A typical plain film X-ray involves radiation dose of 0.01 to 0.15&amp;nbsp;mGy, while a typical CT can involve 10–20&amp;nbsp;mGy for specific organs, and can go up to 80&amp;nbsp;mGy for certain specialized CT scans.&lt;ref name=&quot;crfdr&quot;&gt;{{Cite journal |vauthors=Hall EJ, Brenner DJ |date=May 2008 |title=Cancer risks from diagnostic radiology |journal=The British Journal of Radiology |volume=81 |issue=965 |pages=362–78 |doi=10.1259/bjr/01948454 |pmid=18440940 |s2cid=23348032}}&lt;/ref&gt;

For purposes of comparison, the world average dose rate from naturally occurring sources of [[w:background radiation|background radiation]] is 2.4&amp;nbsp;[[w:mSv|mSv]] per year, equal for practical purposes in this application to 2.4&amp;nbsp;mGy per year.&lt;ref name=&quot;background&quot;&gt;{{Cite journal |vauthors=Cuttler JM, Pollycove M |year=2009 |title=Nuclear energy and health: and the benefits of low-dose radiation hormesis |journal=Dose-Response |volume=7 |issue=1 |pages=52–89 |doi=10.2203/dose-response.08-024.Cuttler |pmc=2664640 |pmid=19343116}}&lt;/ref&gt; While there is some variation, most people (99%) received less than 7&amp;nbsp;mSv per year as background radiation.&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=qCebxPjdSBUC&amp;pg=PA164 |title=A half century of health physics |publisher=Lippincott Williams &amp; Wilkins |year=2005 |isbn=978-0-7817-6934-1 |editor-last=Michael T. Ryan |location=Baltimore, Md. |page=164 |editor-last2=Poston, John W.}}&lt;/ref&gt; Medical imaging as of 2007 accounted for half of the radiation exposure of those in the United States with CT scans making up two thirds of this amount.&lt;ref name=&quot;Furlow2010&quot; /&gt; In the United Kingdom it accounts for 15% of radiation exposure.&lt;ref name=&quot;Risk2011&quot; /&gt; The average radiation dose from medical sources is ≈0.6&amp;nbsp;mSv per person globally as of 2007.&lt;ref name=&quot;Furlow2010&quot; /&gt; Those in the nuclear industry in the United States are limited to doses of 50&amp;nbsp;mSv a year and 100&amp;nbsp;mSv every 5 years.&lt;ref name=&quot;Furlow2010&quot; /&gt;

==== Radiation dose units ====
The radiation dose reported in the [[w:Gray (unit)|gray or mGy]] unit is proportional to the amount of energy that the irradiated body part is expected to absorb, and the physical effect (such as DNA [[w:double strand breaks|double strand breaks]]) on the cells' chemical bonds by X-ray radiation is proportional to that energy.&lt;ref&gt;{{Cite journal |vauthors=Polo SE, Jackson SP |date=March 2011 |title=Dynamics of DNA damage response proteins at DNA breaks: a focus on protein modifications |journal=Genes Dev. |volume=25 |issue=5 |pages=409–33 |doi=10.1101/gad.2021311 |pmc=3049283 |pmid=21363960}}&lt;/ref&gt;

The [[w:sievert|sievert]] unit is used in the report of the [[w:effective dose (radiation)|effective dose]]. The sievert unit, in the context of CT scans, does not correspond to the actual radiation dose that the scanned body part absorbs but to another radiation dose of another scenario, the whole body absorbing the other radiation dose and the other radiation dose being of a magnitude, estimated to have the same probability to induce cancer as the CT scan.&lt;ref&gt;{{Cite journal|last=McCollough|first=Cynthia|last2=Cody|first2=Dianna|last3=Edyvean|first3=Sue|last4=Geise|first4=Rich|last5=Gould|first5=Bob|last6=Keat|first6=Nicholas|last7=Huda|first7=Walter|last8=Judy|first8=Phil|last9=Kalender|first9=Willi|date=2008-01|title=The Measurement, Reporting, and Management of Radiation Dose in CT|url=https://doi.org/10.37206/97|doi=10.37206/97}}&lt;/ref&gt; Thus, as is shown in the table above, the actual radiation that is absorbed by a scanned body part is often much larger than the effective dose suggests. A specific measure, termed the [[w:computed tomography dose index|computed tomography dose index]] (CTDI), is commonly used as an estimate of the radiation absorbed dose for tissue within the scan region, and is automatically computed by medical CT scanners.&lt;ref&gt;{{Cite journal |vauthors=Hill B, Venning AJ, Baldock C |year=2005 |title=A preliminary study of the novel application of normoxic polymer gel dosimeters for the measurement of CTDI on diagnostic X-ray CT scanners |journal=Medical Physics |volume=32 |issue=6 |pages=1589–1597 |bibcode=2005MedPh..32.1589H |doi=10.1118/1.1925181 |pmid=16013718}}&lt;/ref&gt; It is usually expressed in units of milligray (mGy). CTDI is further divided into CTDI&lt;sub&gt;100&lt;/sub&gt;, CTDI&lt;sub&gt;w&lt;/sub&gt; and CTDI&lt;sub&gt;vol&lt;/sub&gt;. CTDI is expressed mathematically as:&lt;ref&gt;{{Cite journal|last=Leon|first=Stephanie M.|last2=Kobistek|first2=Robert J.|last3=Olguin|first3=Edmond A.|last4=Zhang|first4=Zhongwei|last5=Barreto|first5=Izabella L.|last6=Schwarz|first6=Bryan C.|date=2020-08|title=The helically‐acquired CTDI vol as an alternative to traditional methodology|url=https://aapm.onlinelibrary.wiley.com/doi/10.1002/acm2.12944|journal=Journal of Applied Clinical Medical Physics|language=en|volume=21|issue=8|pages=263–271|doi=10.1002/acm2.12944|issn=1526-9914|pmc=PMC7484853|pmid=32519415}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Mekonin|first=Tadelech S.|last2=Deressu|first2=Tilahun T.|date=2022-07|title=Computed Dosimeter Dose Index on a 16-Slice Computed Tomography Scanner|url=http://journals.sagepub.com/doi/10.1177/15593258221119299|journal=Dose-Response|language=en|volume=20|issue=3|pages=155932582211192|doi=10.1177/15593258221119299|issn=1559-3258|pmc=PMC9403463|pmid=36034103}}&lt;/ref&gt;
{| class=&quot;sortable wikitable&quot; style=&quot;float: right; margin-left:15px; text-align:center&quot;
|+Computed Tomography Dose Index
!Radiation exposure Index
!Formula
!
|-
|Computed tomography dose index
|&lt;math display=&quot;block&quot;&gt;CTDI=\frac{1}{nT}\int_{-z}^{+z}{D(z)\text{d}z}&lt;/math&gt;
|Where &lt;math&gt;n&lt;/math&gt; represent the number of slices obtained during a single axial rotation, &lt;math&gt;T&lt;/math&gt; denote the width of an individual acquired slice, and &lt;math&gt;D(z)&lt;/math&gt; represent the radiation dose recorded at position &lt;math&gt;z&lt;/math&gt; along the primary axis of the scanner.&lt;ref name=&quot;auto3&quot;&gt;{{Cite journal|last=Teeuwisse|first=W M|last2=Geleijns|first2=J|last3=Broerse|first3=J J|last4=Obermann|first4=W R|last5=van Persijn van Meerten|first5=E L|date=2001-08|title=Patient and staff dose during CT guided biopsy, drainage and coagulation|url=http://www.birpublications.org/doi/10.1259/bjr.74.884.740720|journal=The British Journal of Radiology|language=en|volume=74|issue=884|pages=720–726|doi=10.1259/bjr.74.884.740720|issn=0007-1285}}&lt;/ref&gt;
|-
|Computed tomography dose index 100
|&lt;math&gt;CTDI_{100}=\frac{1}{nT}\int_{-50 mm}^{50 mm}{D(z)\text{d}z}&lt;/math&gt;.
|
|-
|Computed tomography dose index weighted
|&lt;math&gt;CTDI_w=\frac{1}{3} CTDI_{100}^{central} + \frac{2}{3} CTDI_{100}^{peripheral}&lt;/math&gt;
|&lt;math&gt;CTDI^{central}&lt;/math&gt; is measured at center and &lt;math&gt;CTDI^{peripheral}&lt;/math&gt; is measured at the periphery.&lt;ref name=&quot;auto3&quot;/&gt;
|-
|Computed tomography dose index volume
|&lt;math&gt;CTDI_{vol}=\frac{CTDI_{w}}{P}&lt;/math&gt;
|
|}

Another important parameter in assessing radiation dose in CT is the Dose Length Product (DLP) which is calculated as the product of CTDI&lt;sub&gt;vol&lt;/sub&gt; and the scan length expressed as mGy*cm. It provides an indication of the overall dose output, taking into consideration the length of the scan. However, DLP does not account for the size of the patient and is not a direct measure of absorbed dose or effective dose.To account for variations in patient size, an additional metric, Size-Specific Dose Estimate (SSDE), is introduced. SSDE, measured in mGy, takes into account the size of the patient, providing a more estimate of absorbed dose. It focuses on accounting for variations in patient size when estimating the absorbed dose during a CT examination

The [[w:equivalent dose|equivalent dose]] is the effective dose of a case, in which the whole body would actually absorb the same radiation dose, and the sievert unit is used in its report. In the case of non-uniform radiation, or radiation given to only part of the body, which is common for CT examinations, using the local equivalent dose alone would overstate the biological risks to the entire organism.&lt;ref&gt;{{Cite book |last1=Issa |first1=Ziad F. |title=Clinical Arrhythmology and Electrophysiology |last2=Miller |first2=John M. |last3=Zipes |first3=Douglas P. |date=2019-01-01 |publisher=Elsevier |isbn=978-0-323-52356-1 |pages=1042–1067 |language=en |chapter=Complications of Catheter Ablation of Cardiac Arrhythmias |doi=10.1016/b978-0-323-52356-1.00032-3}}&lt;/ref&gt;&lt;ref&gt;{{Cite web |title=Absorbed, Equivalent, and Effective Dose – ICRPaedia |url=http://icrpaedia.org/Absorbed,_Equivalent,_and_Effective_Dose |access-date=2021-03-21 |website=icrpaedia.org}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last=Materials |first=National Research Council (US) Committee on Evaluation of EPA Guidelines for Exposure to Naturally Occurring Radioactive |url=https://www.ncbi.nlm.nih.gov/books/NBK230653/ |title=Radiation Quantities and Units, Definitions, Acronyms |date=1999 |publisher=National Academies Press (US) |language=en}}&lt;/ref&gt;

==== Effects of radiation ====
Most adverse health effects of radiation exposure may be grouped in two general categories:
*deterministic effects (harmful tissue reactions) due in large part to the killing/malfunction of cells following high doses;&lt;ref&gt;{{Cite book |last1=Pua |first1=Bradley B. |url=https://books.google.com/books?id=7fpyDwAAQBAJ&amp;q=deterministic+effects&amp;pg=PA53 |title=Interventional Radiology: Fundamentals of Clinical Practice |last2=Covey |first2=Anne M. |last3=Madoff |first3=David C. |date=2018-12-03 |publisher=Oxford University Press |isbn=978-0-19-027624-9 |language=en}}&lt;/ref&gt;
*stochastic effects, i.e., cancer and heritable effects involving either cancer development in exposed individuals owing to mutation of somatic cells or heritable disease in their offspring owing to mutation of reproductive (germ) cells.&lt;ref&gt;Paragraph 55 in: {{Cite web |title=The 2007 Recommendations of the International Commission on Radiological Protection |url=http://www.icrp.org/publication.asp?id=ICRP%20Publication%20103 |url-status=live |archive-url=https://web.archive.org/web/20121116084754/http://www.icrp.org/publication.asp?id=ICRP+Publication+103 |archive-date=2012-11-16 |website=[[International Commission on Radiological Protection]]}} Ann. ICRP 37 (2-4)&lt;/ref&gt;

The added lifetime risk of developing cancer by a single abdominal CT of 8 mSv is estimated to be 0.05%, or 1 one in 2,000.&lt;ref&gt;{{Cite web |date=March 2013 |title=Do CT scans cause cancer? |url=https://www.health.harvard.edu/staying-healthy/do-ct-scans-cause-cancer |url-status=dead |archive-url=https://web.archive.org/web/20171209152338/https://www.health.harvard.edu/staying-healthy/do-ct-scans-cause-cancer |archive-date=2017-12-09 |access-date=2017-12-09 |website=[[Harvard Medical School]]}}&lt;/ref&gt;

Because of increased susceptibility of fetuses to radiation exposure, the radiation dosage of a CT scan is an important consideration in the choice of [[w:medical imaging in pregnancy|medical imaging in pregnancy]].&lt;ref&gt;{{Cite web |last=CDC |date=2020-06-05 |title=Radiation and pregnancy: A fact sheet for clinicians |url=https://www.cdc.gov/nceh/radiation/emergencies/prenatalphysician.htm |access-date=2021-03-21 |website=Centers for Disease Control and Prevention |language=en-us}}&lt;/ref&gt;&lt;ref&gt;{{Citation |last1=Yoon |first1=Ilsup |title=Radiation Exposure In Pregnancy |date=2021 |url=http://www.ncbi.nlm.nih.gov/books/NBK551690/ |work=StatPearls |place=Treasure Island (FL) |publisher=StatPearls Publishing |pmid=31869154 |access-date=2021-03-21 |last2=Slesinger |first2=Todd L.}}&lt;/ref&gt;

==== Excess doses ====
In October, 2009, the US [[w:Food and Drug Administration|Food and Drug Administration]] (FDA) initiated an investigation of brain perfusion CT (PCT) scans, based on [[w:radiation burn|radiation burn]]s caused by incorrect settings at one particular facility for this particular type of CT scan. Over 256 patients were exposed to radiations for over 18-month period. Over 40% of them lost patches of hair, and prompted the editorial to call for increased CT quality assurance programs. It was noted that &quot;while unnecessary radiation exposure should be avoided, a medically needed CT scan obtained with appropriate acquisition parameter has benefits that outweigh the radiation risks.&quot;&lt;ref name=&quot;Furlow2010&quot;&gt;{{Cite book |last=Whaites |first=Eric |url=https://books.google.com/books?id=qdOSDdETuxcC&amp;q=Typical+effective+dose&amp;pg=PA27 |title=Radiography and Radiology for Dental Care Professionals E-Book |date=2008-10-10 |publisher=Elsevier Health Sciences |isbn=978-0-7020-4799-2 |pages=25 |language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal |vauthors=Wintermark M, Lev MH |date=January 2010 |title=FDA investigates the safety of brain perfusion CT |journal=AJNR Am J Neuroradiol |volume=31 |issue=1 |pages=2–3 |doi=10.3174/ajnr.A1967 |pmc=7964089 |pmid=19892810 |doi-access=free}}&lt;/ref&gt; Similar problems have been reported at other centers.&lt;ref name=&quot;Furlow2010&quot; /&gt; These incidents are believed to be due to [[w:human error|human error]].&lt;ref name=&quot;Furlow2010&quot; /&gt;

=== Cancer ===
The [[w:ionizing radiation|radiation]] used in CT scans can damage body cells, including [[w:DNA molecule|DNA molecule]]s, which can lead to [[w:radiation-induced cancer|radiation-induced cancer]].&lt;ref name=&quot;Brenner2007&quot; /&gt; The radiation doses received from CT scans is variable.  Compared to the lowest dose x-ray techniques, CT scans can have 100 to 1,000 times higher dose than conventional X-rays.&lt;ref name=&quot;Redberg&quot;&gt;Redberg, Rita F., and Smith-Bindman, Rebecca. [https://www.nytimes.com/2014/01/31/opinion/we-are-giving-ourselves-cancer.html &quot;We Are Giving Ourselves Cancer&quot;] {{webarchive|url=https://web.archive.org/web/20170706163542/https://www.nytimes.com/2014/01/31/opinion/we-are-giving-ourselves-cancer.html?nl=opinion&amp;emc=edit_ty_20140131&amp;_r=0|date=2017-07-06}}, ''New York Times'', January 30, 2014&lt;/ref&gt; However, a lumbar spine x-ray has a similar dose as a head CT.&lt;ref&gt;{{Cite web|url=https://www.fda.gov/Radiation-EmittingProducts/RadiationEmittingProductsandProcedures/MedicalImaging/MedicalX-Rays/ucm115329.htm|title=Medical X-ray Imaging – What are the Radiation Risks from CT?|last=Health|first=Center for Devices and Radiological|website=www.fda.gov|archive-url=https://web.archive.org/web/20131105050317/https://www.fda.gov/Radiation-EmittingProducts/RadiationEmittingProductsandProcedures/MedicalImaging/MedicalX-Rays/ucm115329.htm|archive-date=5 November 2013|access-date=1 May 2018|url-status=live}}&lt;/ref&gt; Articles in the media often exaggerate the relative dose of CT by comparing the lowest-dose x-ray techniques (chest x-ray) with the highest-dose CT techniques. In general, a routine abdominal CT has a radiation dose similar to three years of average [[w:background radiation|background radiation]].&lt;ref&gt;{{Cite web|url=https://www.acr.org/-/media/ACR/Files/Radiology-Safety/Radiation-Safety/Dose-Reference-Card.pdf|title=Patient Safety – Radiation Dose in X-Ray and CT Exams|last=(ACR)|first=[[Radiological Society of North America]] (RSNA) and [[American College of Radiology]]|date=February 2021|website=acr.org|archive-url=https://web.archive.org/web/20210101161039/https://www.acr.org/-/media/ACR/Files/Radiology-Safety/Radiation-Safety/Dose-Reference-Card.pdf|archive-date=1 January 2021|access-date=6 April 2021|url-status=dead}}&lt;/ref&gt;

Studies published in 2020 on 2.5 million patients&lt;ref name=&quot;Patients undergoing recurrent CT sc&quot;&gt;{{Cite journal|last1=Rehani|first1=Madan M.|last2=Yang|first2=Kai|last3=Melick|first3=Emily R.|last4=Heil|first4=John|last5=Šalát|first5=Dušan|last6=Sensakovic|first6=William F.|last7=Liu|first7=Bob|year=2020|title=Patients undergoing recurrent CT scans: assessing the magnitude|journal=European Radiology|volume=30|issue=4|pages=1828–1836|doi=10.1007/s00330-019-06523-y|pmid=31792585|s2cid=208520824}}&lt;/ref&gt; and 3.2 million patients&lt;ref name=&quot;Multinational data on cumulative ra&quot;&gt;{{Cite journal|last1=Brambilla|first1=Marco|last2=Vassileva|first2=Jenia|last3=Kuchcinska|first3=Agnieszka|last4=Rehani|first4=Madan M.|year=2020|title=Multinational data on cumulative radiation exposure of patients from recurrent radiological procedures: call for action|journal=European Radiology|volume=30|issue=5|pages=2493–2501|doi=10.1007/s00330-019-06528-7|pmid=31792583|s2cid=208520544}}&lt;/ref&gt; have drawn attention to high cumulative doses of more than 100 mSv to patients undergoing recurrent CT scans within a short time span of 1 to 5 years.

Some experts note that CT scans are known to be &quot;overused,&quot; and &quot;there is distressingly little evidence of better health outcomes associated with the current high rate of scans.&quot;&lt;ref name=&quot;Redberg&quot; /&gt; On the other hand, a recent paper analyzing the data of patients who received high [[w:cumulative dose|cumulative dose]]s showed a high degree of appropriate use.&lt;ref name=&quot;Patients undergoing recurrent CT ex&quot;&gt;{{Cite journal|last1=Rehani|first1=Madan M.|last2=Melick|first2=Emily R.|last3=Alvi|first3=Raza M.|last4=Doda Khera|first4=Ruhani|last5=Batool-Anwar|first5=Salma|last6=Neilan|first6=Tomas G.|last7=Bettmann|first7=Michael|year=2020|title=Patients undergoing recurrent CT exams: assessment of patients with non-malignant diseases, reasons for imaging and imaging appropriateness|journal=European Radiology|volume=30|issue=4|pages=1839–1846|doi=10.1007/s00330-019-06551-8|pmid=31792584|s2cid=208520463}}&lt;/ref&gt; This creates an important issue of cancer risk to these patients. Moreover, a highly significant finding that was previously unreported is that some patients received &gt;100 mSv dose from CT scans in  a single day,&lt;ref name=&quot;Patients undergoing recurrent CT sc&quot; /&gt; which counteracts existing criticisms some investigators may have on the effects of protracted versus acute exposure.

Early estimates of harm from CT are partly based on similar radiation exposures experienced by those present during the [[w:atomic bomb|atomic bomb]] explosions in Japan after the [[w:World War II|Second World War]] and those of [[w:nuclear industry|nuclear industry]] workers.&lt;ref name=&quot;Brenner2007&quot; /&gt; Some experts project that in the future, between three and five percent of all cancers would result from medical imaging.&lt;ref name=&quot;Redberg&quot; /&gt;

An Australian study of 10.9&amp;nbsp;million people reported that the increased incidence of cancer after CT scan exposure in this cohort was mostly due to irradiation. In this group, one in every 1,800 CT scans was followed by an excess cancer. If the lifetime risk of developing cancer is 40% then the absolute risk rises to 40.05% after a CT.&lt;ref name=&quot;MathewsForsythe2013&quot;&gt;{{Cite journal|last1=Mathews|first1=J. D.|last2=Forsythe|first2=A. V.|last3=Brady|first3=Z.|last4=Butler|first4=M. W.|last5=Goergen|first5=S. K.|last6=Byrnes|first6=G. B.|last7=Giles|first7=G. G.|last8=Wallace|first8=A. B.|last9=Anderson|first9=P. R.|year=2013|title=Cancer risk in 680 000 people exposed to computed tomography scans in childhood or adolescence: data linkage study of 11 million Australians|journal=BMJ|volume=346|issue=may21 1|pages=f2360|doi=10.1136/bmj.f2360|issn=1756-1833|pmc=3660619|pmid=23694687|last10=Guiver|first10=T. A.|last11=McGale|first11=P.|last12=Cain|first12=T. M.|last13=Dowty|first13=J. G.|last14=Bickerstaffe|first14=A. C.|last15=Darby|first15=S. C.}}&lt;/ref&gt;&lt;ref name=&quot;SasieniShelton2011&quot;&gt;{{Cite journal|last1=Sasieni|first1=P D|last2=Shelton|first2=J|last3=Ormiston-Smith|first3=N|last4=Thomson|first4=C S|last5=Silcocks|first5=P B|year=2011|title=What is the lifetime risk of developing cancer?: the effect of adjusting for multiple primaries|journal=British Journal of Cancer|volume=105|issue=3|pages=460–465|doi=10.1038/bjc.2011.250|issn=0007-0920|pmc=3172907|pmid=21772332}}&lt;/ref&gt;

Some studies have shown that publications indicating an increased risk of cancer from typical doses of body CT scans are plagued with serious methodological limitations and several highly improbable results,&lt;ref&gt;{{Cite journal|last1=Eckel|first1=Laurence J.|last2=Fletcher|first2=Joel G.|last3=Bushberg|first3=Jerrold T.|last4=McCollough|first4=Cynthia H.|date=2015-10-01|title=Answers to Common Questions About the Use and Safety of CT Scans|url=https://www.mayoclinicproceedings.org/article/S0025-6196(15)00591-1/fulltext|journal=Mayo Clinic Proceedings|language=en|volume=90|issue=10|pages=1380–1392|doi=10.1016/j.mayocp.2015.07.011|issn=0025-6196|pmid=26434964|doi-access=free}}&lt;/ref&gt; concluding that no evidence indicates such low doses cause any long-term harm.&lt;ref&gt;{{Cite web|url=https://www.sciencedaily.com/releases/2015/10/151005151507.htm|title=Expert opinion: Are CT scans safe?|website=ScienceDaily|language=en|access-date=2019-03-14}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=McCollough|first1=Cynthia H.|last2=Bushberg|first2=Jerrold T.|last3=Fletcher|first3=Joel G.|last4=Eckel|first4=Laurence J.|date=2015-10-01|title=Answers to Common Questions About the Use and Safety of CT Scans|url=https://www.mayoclinicproceedings.org/article/S0025-6196(15)00591-1/abstract|journal=Mayo Clinic Proceedings|language=English|volume=90|issue=10|pages=1380–1392|doi=10.1016/j.mayocp.2015.07.011|issn=0025-6196|pmid=26434964|doi-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.medicalnewstoday.com/articles/306067.php|title=No evidence that CT scans, X-rays cause cancer|date=4 February 2016|website=Medical News Today|language=en|access-date=2019-03-14}}&lt;/ref&gt;

One study estimated that as many as 0.4% of cancers in the United States resulted from CT scans, and that this may have increased to as much as 1.5 to 2% based on the rate of CT use in 2007.&lt;ref name=&quot;Brenner2007&quot; /&gt; Others dispute this estimate,&lt;ref&gt;{{Cite journal|last1=Kalra|first1=Mannudeep K.|last2=Maher|first2=Michael M.|last3=Rizzo|first3=Stefania|last4=Kanarek|first4=David|last5=Shephard|first5=Jo-Anne O.|date=April 2004|title=Radiation exposure from Chest CT: Issues and Strategies|journal=Journal of Korean Medical Science|volume=19|issue=2|pages=159–166|doi=10.3346/jkms.2004.19.2.159|issn=1011-8934|pmc=2822293|pmid=15082885}}&lt;/ref&gt; as there is no consensus that the low levels of radiation used in CT scans cause damage. Lower radiation doses are used in many cases, such as in the investigation of renal colic.&lt;ref&gt;{{Cite journal|last1=Rob|first1=S.|last2=Bryant|first2=T.|last3=Wilson|first3=I.|last4=Somani|first4=B.K.|year=2017|title=Ultra-low-dose, low-dose, and standard-dose CT of the kidney, ureters, and bladder: is there a difference? Results from a systematic review of the literature|journal=Clinical Radiology|volume=72|issue=1|pages=11–15|doi=10.1016/j.crad.2016.10.005|pmid=27810168}}&lt;/ref&gt;

A person's age plays a significant role in the subsequent risk of cancer.&lt;ref name=&quot;Furlow2010&quot; /&gt; Estimated lifetime cancer mortality risks from an abdominal CT of a one-year-old is 0.1%, or 1:1000 scans.&lt;ref name=&quot;Furlow2010&quot; /&gt; The risk for someone who is 40 years old is half that of someone who is 20 years old with substantially less risk in the elderly.&lt;ref name=&quot;Furlow2010&quot; /&gt; The [[w:International Commission on Radiological Protection|International Commission on Radiological Protection]] estimates that the risk to a fetus being exposed to 10 [[w:mGy|mGy]] (a unit of radiation exposure) increases the rate of cancer before 20 years of age from 0.03% to 0.04% (for reference a CT pulmonary angiogram exposes a fetus to 4&amp;nbsp;mGy).&lt;ref name=&quot;Risk2011&quot; /&gt; A 2012 review did not find an association between medical radiation and cancer risk in children noting however the existence of limitations in the evidences over which the review is based.&lt;ref&gt;{{Cite journal|date=January 2012|title=[Diagnostic radiation exposure in children and cancer risk: current knowledge and perspectives]|journal=Archives de Pédiatrie|volume=19|issue=1|pages=64–73|doi=10.1016/j.arcped.2011.10.023|pmid=22130615|vauthors=Baysson H, Etard C, Brisse HJ, Bernier MO}}&lt;/ref&gt;

CT scans can be performed with different settings for lower exposure in children with most manufacturers of CT scans as of 2007 having this function built in.&lt;ref name=&quot;Semelka2007&quot; /&gt; Furthermore, certain conditions can require children to be exposed to multiple CT scans.&lt;ref name=&quot;Brenner2007&quot; /&gt; Current evidence suggests informing parents of the risks of pediatric CT scanning.&lt;ref name=&quot;pmid17646450&quot;&gt;{{Cite journal|date=August 2007|title=Informing parents about CT radiation exposure in children: it's OK to tell them|journal=Am J Roentgenol|volume=189|issue=2|pages=271–5|doi=10.2214/AJR.07.2248|pmid=17646450|vauthors=Larson DB, Rader SB, Forman HP, Fenton LZ|s2cid=25020619}}&lt;/ref&gt;

=== Risks vs benefits ===
The decision to request a CT scan involves an evaluation of potential benefits and associated risks. CT scans are invaluable for precise diagnoses, efficient treatment planning, and time-saving advantages across various medical contexts. While CT scans offer detailed anatomical insights for disease diagnosis and improved patient management, the use of ionizing radiation and contrast media raises concerns about potential harm and adverse effects. Every CT procedure must be justified, and the potential benefits should outweigh the associated risks. Preferential use of alternative, non-ionizing modalities such as ultrasound or MRI should be sought when diagnostically significant imaging can be obtained from them without compromising accuracy. In instances where a CT examination is deemed indispensable, diligent measures should be undertaken to optimize the procedure, ensuring the utilization of the minimum radiation dose essential for achieving diagnostically accurate scans. Adherence to the ALARA  guidelines becomes important to mitigate unnecessary radiation exposure, prioritizing the well-being of the patient.

== History ==
The history of X-ray computed tomography goes back to at least 1917 with the mathematical theory of the [[w:Radon transform|Radon transform]].&lt;ref name=&quot;Radon1917&quot;&gt;{{Cite book |last1=Thomas |first1=Adrian M. K. |url=https://books.google.com/books?id=zgezC3Osm8QC&amp;q=info:6gaBWGuVV0UJ:scholar.google.com/&amp;pg=PA5 |title=Classic Papers in Modern Diagnostic Radiology |last2=Banerjee |first2=Arpan K. |last3=Busch |first3=Uwe |date=2005-12-05 |publisher=Springer Science &amp; Business Media |isbn=978-3-540-26988-5 |language=en}}&lt;/ref&gt;&lt;ref name=&quot;pmid 18244009&quot;&gt;{{Cite journal |last=Radon J |date=1 December 1986 |title=On the determination of functions from their integral values along certain manifolds |journal=IEEE Transactions on Medical Imaging |volume=5 |issue=4 |pages=170–176 |doi=10.1109/TMI.1986.4307775 |pmid=18244009 |s2cid=26553287}}&lt;/ref&gt; In October 1963, [[w:William H. Oldendorf|William H. Oldendorf]] received a U.S. patent for a &quot;radiant energy apparatus for investigating selected areas of interior objects obscured by dense material&quot;.&lt;ref name=&quot;Oldendorf1978&quot;&gt;{{Cite journal |last=Oldendorf WH |year=1978 |title=The quest for an image of brain: a brief historical and technical review of brain imaging techniques |journal=Neurology |volume=28 |issue=6 |pages=517–33 |doi=10.1212/wnl.28.6.517 |pmid=306588 |s2cid=42007208}}&lt;/ref&gt; The first commercially viable CT scanner was invented by [[w:Godfrey Hounsfield|Godfrey Hounsfield]] in 1972.&lt;ref name=&quot;Richmond2004&quot;&gt;{{Cite journal |last=Richmond |first=Caroline |year=2004 |title=Obituary – Sir Godfrey Hounsfield |journal=BMJ |volume=329 |issue=7467 |pages=687 |doi=10.1136/bmj.329.7467.687 |pmc=517662}}&lt;/ref&gt; The 1979 [[w:Nobel Prize in Physiology or Medicine|Nobel Prize in Physiology or Medicine]] was awarded jointly to South African-American physicist [[w:Allan MacLeod Cormack|Allan MacLeod Cormack]] and British electrical engineer [[w:Godfrey Hounsfield|Godfrey Hounsfield]] &quot;for the development of computer-assisted tomography&quot;.&lt;ref&gt;{{Cite journal|last=Di Chiro|first=Giovanni|last2=Brooks|first2=Rodney A.|date=1979-11-30|title=The 1979 Nobel Prize in Physiology or Medicine|url=http://dx.doi.org/10.1126/science.386516|journal=Science|volume=206|issue=4422|pages=1060–1062|doi=10.1126/science.386516|issn=0036-8075}}&lt;/ref&gt;

=== Etymology ===
The word &quot;tomography&quot; is derived from the [[w:Ancient Greek|Greek]] ''tome'' (slice) and ''graphein'' (to write).&lt;ref&gt;{{Cite book |last=[[Frank Natterer]] |title=The Mathematics of Computerized Tomography (Classics in Applied Mathematics) |publisher=Society for Industrial and Applied Mathematics |year=2001 |isbn=978-0-89871-493-7 |pages=8}}&lt;/ref&gt; Computed tomography was originally known as the &quot;EMI scan&quot; as it was developed in the early 1970s at a research branch of [[w:EMI|EMI]], a company best known today for its music and recording business.&lt;ref&gt;{{Cite book |last=Sperry |first=Len |url=https://books.google.com/books?id=NzgVCwAAQBAJ&amp;q=Computed+tomography+was+originally+known+as+the+%22EMI+scan%22&amp;pg=PA259 |title=Mental Health and Mental Disorders: An Encyclopedia of Conditions, Treatments, and Well-Being [3 volumes]: An Encyclopedia of Conditions, Treatments, and Well-Being |date=2015-12-14 |publisher=ABC-CLIO |isbn=978-1-4408-0383-3 |page=259 |language=en}}&lt;/ref&gt; It was later known as ''computed axial tomography'' (''CAT'' or ''CT scan'') and ''body section röntgenography''.&lt;ref&gt;{{Cite journal |last=Hounsfield |first=G. N. |date=1977 |title=The E.M.I. Scanner |journal=Proceedings of the Royal Society of London. Series B, Biological Sciences |volume=195 |issue=1119 |pages=281–289 |bibcode=1977RSPSB.195..281H |doi=10.1098/rspb.1977.0008 |issn=0080-4649 |jstor=77187 |pmid=13396 |s2cid=34734270}}&lt;/ref&gt; The term &quot;CAT scan&quot; is no longer used because current CT scans enable for multiplanar reconstructions. This makes &quot;CT scan&quot; the most appropriate term, which is used by [[w:radiologist|radiologist]]s in common vernacular as well as in textbooks and scientific papers.&lt;ref&gt;{{Cite web |title=Difference Between CT Scan and CAT Scan {{!}} Difference Between |url=http://www.differencebetween.net/science/health/difference-between-ct-scan-and-cat-scan/ |access-date=2021-03-19 |language=en-US}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=FqDUtcmUG-UC&amp;q=cat+scanner+term+was+used+earlier |title=Conquer Your Headaches |publisher=International Headache Management |year=1994 |isbn=978-0-9636292-5-8 |pages=115}}&lt;/ref&gt; In [[w:Medical Subject Headings|Medical Subject Headings]] (MeSH), &quot;computed axial tomography&quot; was used from 1977 to 1979, but the current indexing explicitly includes &quot;X-ray&quot; in the title.&lt;ref&gt;{{Cite web |title=MeSH Browser |url=https://meshb.nlm.nih.gov/record/ui?ui=D014057 |website=meshb.nlm.nih.gov}}&lt;/ref&gt;

== Society and culture ==
=== Campaigns ===
In response to increased concern by the public and the ongoing progress of best practices, the Alliance for Radiation Safety in Pediatric Imaging was formed within the [[w:Society for Pediatric Radiology|Society for Pediatric Radiology]]. In concert with the [[w:American Society of Radiologic Technologists|American Society of Radiologic Technologists]], the [[w:American College of Radiology|American College of Radiology]] and the [[w:American Association of Physicists in Medicine|American Association of Physicists in Medicine]], the Society for Pediatric Radiology developed and launched the Image Gently Campaign which is designed to maintain high-quality imaging studies while using the lowest doses and best radiation safety practices available on pediatric patients.&lt;ref&gt;{{Cite web |title=Image Gently |url=http://www.pedrad.org/associations/5364/ig/?page=365 |url-status=dead |archive-url=https://web.archive.org/web/20130609063515/http://www.pedrad.org/associations/5364/ig/?page=365 |archive-date=9 June 2013 |access-date=19 July 2013 |publisher=The Alliance for Radiation Safety in Pediatric Imaging}}&lt;/ref&gt; This initiative has been endorsed and applied by a growing list of various professional medical organizations around the world and has received support and assistance from companies that manufacture equipment used in Radiology.

Following upon the success of the ''Image Gently'' campaign, the American College of Radiology, the Radiological Society of North America, the American Association of Physicists in Medicine and the American Society of Radiologic Technologists have launched a similar campaign to address this issue in the adult population called ''Image Wisely''.&lt;ref&gt;{{Cite web |title=Image Wisely |url=http://www.imagewisely.org/ |url-status=dead |archive-url=https://web.archive.org/web/20130721032437/http://imagewisely.org/ |archive-date=21 July 2013 |access-date=19 July 2013 |publisher=Joint Task Force on Adult Radiation Protection}}&lt;/ref&gt;

The [[w:World Health Organization|World Health Organization]] and [[w:International Atomic Energy Agency|International Atomic Energy Agency]] (IAEA) of the United Nations have also been working in this area and have ongoing projects designed to broaden best practices and lower patient radiation dose.&lt;ref&gt;{{Cite web |title=Optimal levels of radiation for patients |url=http://new.paho.org/hq10/index.php?option=com_content&amp;task=view&amp;id=3365&amp;Itemid=2164 |url-status=dead |archive-url=https://web.archive.org/web/20130525051814/http://new.paho.org/hq10/index.php?option=com_content&amp;task=view&amp;id=3365&amp;Itemid=2164 |archive-date=25 May 2013 |access-date=19 July 2013 |publisher=World Health Organization}}&lt;/ref&gt;

=== Prevalence ===
&lt;div style=&quot;float:right; width:23em; height:20em; overflow:auto; border:0px&quot;&gt;
{|class=&quot;wikitable&quot;
|+{{nowrap|Number of  CT scanners by country (OECD)}}&lt;br /&gt;as of 2017&lt;ref&gt;{{Cite web |title=Computed tomography (CT) scanners |url=https://data.oecd.org/healtheqt/computed-tomography-ct-scanners.htm |publisher=OECD}}&lt;/ref&gt;&lt;br /&gt;(per million population)
!Country !! Value
|-
|{{flagcountry| JPN}} || 111.49
|-
|{{flagcountry| AUS}} || 64.35
|-
|{{flagcountry| ISL}} || 43.68
|-
|{{flagcountry| USA}} || 42.64
|-
|{{flagcountry| DNK}} || 39.72
|-
|{{flagcountry| CHE}} || 39.28
|-
|{{flagcountry| LVA}} || 39.13
|-
|{{flagcountry| KOR}} || 38.18
|-
|{{flagcountry| DEU}} || 35.13
|-
|{{flagcountry| ITA}} || 34.71
|-
|{{flagcountry| GRC}} || 34.22
|-
|{{flagcountry| AUT}} || 28.64
|-
|{{flagcountry| FIN}} || 24.51
|-
|{{flagcountry| CHL}} || 24.27
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|{{flagcountry| IRL}} || 19.14
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|{{flagcountry| ESP}} || 18.59
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|{{flagcountry| EST}} || 18.22
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|{{flagcountry| FRA}} || 17.36
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|{{flagcountry| SVK}} || 17.28
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|{{flagcountry| POL}} || 16.88
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|{{flagcountry| CZE}} || 15.76
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|{{flagcountry| CAN}} || 15.28
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|{{flagcountry| SVN}} || 15.00
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|{{flagcountry| TUR}} || 14.77
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&lt;/div&gt;
Use of CT has increased dramatically over the last two decades.&lt;ref name=&quot;Smith2009&quot; /&gt; An estimated 72&amp;nbsp;million scans were performed in the United States in 2007,&lt;ref name=&quot;Berrington2009&quot; /&gt; accounting for close to half of the total per-capita dose rate from radiologic and nuclear medicine procedures.&lt;ref&gt;{{Cite journal |last1=Fred A. Mettler Jr |last2=Mythreyi Bhargavan |last3=Keith Faulkner |last4=Debbie B. Gilley |last5=Joel E. Gray |last6=Geoffrey S. Ibbott |last7=Jill A. Lipoti |last8=Mahadevappa Mahesh |last9=John L. McCrohan |last10=Michael G. Stabin |last11=Bruce R. Thomadsen |last12=Terry T. Yoshizumi |year=2009 |title=Radiologic and Nuclear Medicine Studies in the United States and Worldwide: Frequency, Radiation Dose, and Comparison with Other Radiation Sources — 1950-2007 |journal=Radiology |volume=253 |pages=520–531 |doi=10.1148/radiol.2532082010 |pmid=19789227 |number=2}}&lt;/ref&gt; Of the CT scans, six to eleven percent are done in children,&lt;ref name=&quot;Risk2011&quot; /&gt; an increase of seven to eightfold from 1980.&lt;ref name=&quot;Furlow2010&quot; /&gt; Similar increases have been seen in Europe and Asia.&lt;ref name=&quot;Furlow2010&quot; /&gt;  In Calgary, Canada, 12.1% of people who present to the emergency with an urgent complaint received a CT scan, most commonly either of the head or of the abdomen. The percentage who received CT, however, varied markedly by the [[w:emergency physician|emergency physician]] who saw them from 1.8% to 25%.&lt;ref&gt;{{Cite journal |last=Andrew Skelly |date=Aug 3, 2010 |title=CT ordering all over the map |journal=The Medical Post}}&lt;/ref&gt; In the emergency department in the United States, CT or [[w:Magnetic resonance imaging|MRI]] imaging is done in 15% of people who present with [[w:injuries|injuries]] as of 2007 (up from 6% in 1998).&lt;ref&gt;{{Cite journal |vauthors=Korley FK, Pham JC, Kirsch TD |date=October 2010 |title=Use of advanced radiology during visits to US emergency departments for injury-related conditions, 1998–2007 |journal=JAMA |volume=304 |issue=13 |pages=1465–71 |doi=10.1001/jama.2010.1408 |pmid=20924012 |doi-access=free}}&lt;/ref&gt;

The increased use of CT scans has been the greatest in two fields: screening of adults (screening CT of the lung in smokers, virtual colonoscopy, CT cardiac screening, and whole-body CT in asymptomatic patients) and CT imaging of children. Shortening of the scanning time to around 1 second, eliminating the strict need for the subject to remain still or be sedated, is one of the main reasons for the large increase in the pediatric population (especially for the diagnosis of [[w:appendicitis|appendicitis]]).&lt;ref name=&quot;Brenner2007&quot; /&gt; As of 2007, in the United States a proportion of CT scans are performed unnecessarily.&lt;ref name=&quot;Semelka2007&quot;&gt;{{Cite journal |vauthors=Semelka RC, Armao DM, Elias J, Huda W |date=May 2007 |title=Imaging strategies to reduce the risk of radiation in CT studies, including selective substitution with MRI |journal=J Magn Reson Imaging |volume=25 |issue=5 |pages=900–9 |doi=10.1002/jmri.20895 |pmid=17457809 |s2cid=5788891}}&lt;/ref&gt; Some estimates place this number at 30%.&lt;ref name=&quot;Risk2011&quot; /&gt; There are a number of reasons for this including: legal concerns, financial incentives, and desire by the public.&lt;ref name=&quot;Semelka2007&quot; /&gt; For example, some healthy people avidly pay to receive full-body CT scans as [[w:screening (medicine)|screening]]. In that case, it is not at all clear that the benefits outweigh the risks and costs. Deciding whether and how to treat [[w:incidentaloma|incidentaloma]]s is complex, radiation exposure is not negligible, and the money for the scans involves [[w:opportunity cost|opportunity cost]].&lt;ref name=&quot;Semelka2007&quot; /&gt;

==Additional Information==

=== Acknowledgements ===
I would like to express my gratitude to Prof. Lalit Kumar Gupta (Rayat Bahra University) for their guidance and academic encouragement. 

=== Competing interests ===
The authors declare that they have no competing interests.

=== Ethics statement ===
No ethics statement necessary.

== References ==
{{reflist|refs=
{{Citation |title=Advanced Documentation Methods in Studying Corinthian Black-figure Vase Painting |date=2019 |url=https://www.chnt.at/wp-content/uploads/eBook_CHNT23_Karl.pdf |work=Proceedings of the 23rd International Conference on Cultural Heritage and New Technologies (CHNT23) |publication-place=Vienna, Austria |isbn=978-3-200-06576-5 |access-date=2020-01-09 |last2=Bayer |first2=Paul |author3-link=Hubert Mara |last3=Mara |first3=Hubert |last4=Márton |first4=András |given1=Stephan |surname1=Karl}}&lt;/ref&gt;}}</text>
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| first1 = Peerzada Mohammad
| last1 = Iflaq
| orcid1 = 0009-0005-4796-2375
| affiliation1 = Rayat Bahra University
| correspondence1 = peerzadaiflaq@gmail.com
| affiliations = institutes / affiliations
| et_al = https://en.wikipedia.org/w/index.php?title=CT_scan&amp;oldid=1156317872
| correspondence = email@address.com
| w1 = CT Scan
| from_w1 = true
| journal = WikiJournal of Medicine      &lt;!-- WikiJournal of Medicine, Science, or Humanities --&gt;
| license = &lt;!-- default is CC-BY --&gt;
| abstract = A computed tomography scan (usually abbreviated to CT scan; formerly called computed axial tomography scan or CAT scan) is a [[w:medical imaging|medical imaging]] technique used to obtain detailed internal images of the body. The personnel that perform CT scans are called [[w:radiographer|radiographer]]s or radiology technologists.

CT scanners use a rotating [[w:X-ray tube|X-ray tube]] and a row of detectors placed in a [[w:gantry (medical)|gantry]] to measure X-ray [[w:Attenuation#Radiography|attenuations]] by different tissues inside the body. The multiple [[w:X-ray|X-ray]] measurements taken from different angles are then processed on a computer using [[w:tomographic reconstruction|tomographic reconstruction]] algorithms to produce [[w:Tomography|tomographic]] (cross-sectional) images (virtual &quot;slices&quot;) of a body. CT scan can be used in patients with metallic implants or pacemakers, for whom [[w:magnetic resonance imaging|magnetic resonance imaging]] (MRI) is [[w:Contraindication|contraindicated]].

Compared to conventional [[:w:radiography|X-ray imaging]], CT scans provide detailed cross-sectional information of a specific area under examination, eliminates image superimposition which results in improved diagnostic capabilities. These tomographic images have proven to be valuable for accurate diagnosis and clinicopathological correlation in various [[w:medical condition|medical conditions]].
| keywords = &lt;!-- up to 6 keywords --&gt; CT Scan, Computed Tomography
}}

== Types ==

=== Generations of CT ===
The design and development of CT scanners went through multiple phases, which are collectively referred to as &quot;generations of CT.&quot;
[[File:First Generation CT Scan.svg|109x109px|1st Gen|frameless|right]]
The first-generation CT scanner, developed by Godfrey Hounsfield, also known as EMI scanner operated on the 'translate-rotate' principle. This system employed a pencil X-ray beam and two detectors, facilitating the acquisition of two views in the through-plane direction.&lt;ref&gt;{{Cite journal|last=Hounsfield|first=G. N.|date=1973-12|title=Computerized transverse axial scanning (tomography): Part 1. Description of system|url=http://www.birpublications.org/doi/abs/10.1259/0007-1285-46-552-1016|journal=The British Journal of Radiology|language=en|volume=46|issue=552|pages=1016–1022|doi=10.1259/0007-1285-46-552-1016|issn=0007-1285}}&lt;/ref&gt; The linear translation mechanism enabled the acquisition of 160 rays while the rotational movement captured 180 projections at 1° intervals, resulting in 28,800 rays for linear measurements.&lt;ref&gt;{{Cite journal|last=Ambrose|first=J.|last2=Hounsfield|first2=G.|date=1973-02|title=Computerized transverse axial tomography|url=https://pubmed.ncbi.nlm.nih.gov/4686818/|journal=The British Journal of Radiology|volume=46|issue=542|pages=148–149|issn=0007-1285|pmid=4686818}}&lt;/ref&gt; The CT scanner took about 4 minutes to acquire one slice and 15-20 minuntes to process the data.&lt;ref&gt;{{Cite journal|last=Friedland|first=G W|last2=Thurber|first2=B D|date=1996-12|title=The birth of CT.|url=https://www.ajronline.org/doi/10.2214/ajr.167.6.8956560|journal=American Journal of Roentgenology|language=en|volume=167|issue=6|pages=1365–1370|doi=10.2214/ajr.167.6.8956560|issn=0361-803X}}&lt;/ref&gt;&lt;ref name=&quot;auto7&quot;&gt;{{Cite journal|last=New|first=Paul F. J.|last2=Scott|first2=William R.|last3=Schnur|first3=James A.|last4=Davis|first4=Kenneth R.|last5=Taveras|first5=Juan M.|date=1974-01|title=Computerized Axial Tomography with the EMI Scanner|url=http://pubs.rsna.org/doi/10.1148/110.1.109|journal=Radiology|language=en|volume=110|issue=1|pages=109–123|doi=10.1148/110.1.109|issn=0033-8419}}&lt;/ref&gt; The first prototype model was installed in South London at Atkinson Morley's Hospital, and on October 1, 1971 first patient was scanned.&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Beckmann|first=E C|date=2006-01|title=CT scanning the early days|url=https://academic.oup.com/bjr/article/79/937/5-8/7443496|journal=The British Journal of Radiology|language=en|volume=79|issue=937|pages=5–8|doi=10.1259/bjr/29444122|issn=0007-1285}}&lt;/ref&gt; The first commercial scanner was installed at Mayo Clinic in 1973.&lt;ref name=&quot;auto7&quot;/&gt;
[[File:Second_Generation_CT_Scan.svg|left|frameless|111x111px]]
Following the initial development of the first-generation CT, EMI introduced &quot;CT 1010&quot;  an enhanced scanner in 1975 that eliminated the need for a waterbag. This system also employed &quot;translate-rotate&quot; configuration, but featured an upgraded setup with 8 detectors spanning 3 degrees. This enhancement allowed for a 3-degree rotation increment and required only 60 translations, significantly reducing the scan time to just 1 minute. This innovative design was subsequently referred to as the second-generation CT&lt;ref&gt;{{Cite journal|last=Schulz|first=Raymond A.|last2=Stein|first2=Jay A.|last3=Pelc|first3=Norbert J.|date=2021-10-29|title=How CT happened: the early development of medical computed tomography|url=https://www.spiedigitallibrary.org/journals/journal-of-medical-imaging/volume-8/issue-05/052110/How-CT-happened--the-early-development-of-medical-computed/10.1117/1.JMI.8.5.052110.full|journal=Journal of Medical Imaging|volume=8|issue=05|doi=10.1117/1.JMI.8.5.052110|issn=2329-4302|pmc=PMC8555965|pmid=34729383}}&lt;/ref&gt;. Subsequently, the detector count increased to 30, covering a range of 10 degrees and reducing the scan time to 20 seconds, as seen in the EMI 5000 series.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref&gt;{{Cite book|title=The essential physics of medical imaging|date=2012|publisher=Wolters Kluwer, Lippincott Williams &amp; Wilkins|isbn=978-0-7817-8057-5|editor-last=Bushberg|editor-first=Jerrold T.|edition=3. ed|location=Philadelphia}} p 370&lt;/ref&gt;
[[File:Third generation CT.svg|110x110px|Third generation CT|frameless|right]]
Third generation CT scanners used rotate-rotate configuration i.e. both the tube and the detectors rotated around the patient, employing a wide fan beam x-ray geometry and multiple detectors to collect the data.&lt;ref&gt;{{Cite journal|last=Chen|first=A. C.|last2=Berninger|first2=W. H.|last3=Redington|first3=R. W.|last4=Godbarsen|first4=R.|last5=Barrett|first5=D.|date=1976-12-23|editor-last=Cacak|editor-first=Robert K.|editor2-last=Carson|editor2-first=Paul L.|editor3-last=Dubuque|editor3-first=Gregory|editor4-last=Gray|editor4-first=Joel E.|editor5-last=Hendee|editor5-first=William R.|editor6-last=Rossi|editor6-first=Raymond P.|editor7-last=Haus|editor7-first=Arthur|title=Five-Second Fan Beam CT Scanner|url=http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1226195|pages=294–299|doi=10.1117/12.965426}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Goodenough|first=D. J.|last2=Weaver|first2=K. E.|date=1979-02|title=Overview of Computed Tomography|url=http://ieeexplore.ieee.org/document/4330458/|journal=IEEE Transactions on Nuclear Science|volume=26|issue=1|pages=1661–1667|doi=10.1109/TNS.1979.4330458|issn=0018-9499}}&lt;/ref&gt; In this generation of scanners, a singular detector element malfunction results in the erroneous recording of the corresponding ray in all projections, thereby inducing a ring artifact in the resultant images.&lt;ref&gt;{{Cite book|title=The essential physics of medical imaging|date=2012|publisher=Wolters Kluwer, Lippincott Williams &amp; Wilkins|isbn=978-0-7817-8057-5|editor-last=Bushberg|editor-first=Jerrold T.|edition=3. ed|location=Philadelphia}} p 371.&lt;/ref&gt; EMI 6000 belonged to the third generation of CT scanners.&lt;ref&gt;{{Cite journal|last=Mitchell|first=Se|last2=Clark|first2=Ra|date=1984-04-01|title=A comparison of computed tomography and sonography in choledocholithiasis|url=https://www.ajronline.org/doi/10.2214/ajr.142.4.729|journal=American Journal of Roentgenology|language=en|volume=142|issue=4|pages=729–733|doi=10.2214/ajr.142.4.729|issn=0361-803X}}&lt;/ref&gt;
[[File:Fourth Generation CT Scan.svg|left|frameless|112x112px|Fourth Generation CT Scan]]
The Fourth-generation CT Scanners, initially pioneered by A.S.&amp; E. Corporation,&lt;ref name=&quot;auto4&quot;&gt;{{Cite journal|last=Kak|first=A.C.|date=1979|title=Computerized tomography with X-ray, emission, and ultrasound sources|url=http://ieeexplore.ieee.org/document/1455709/|journal=Proceedings of the IEEE|volume=67|issue=9|pages=1245–1272|doi=10.1109/PROC.1979.11440|issn=0018-9219}}&lt;/ref&gt; employed detectors organized in a fixed ring comprising around 4800 individual detector elements. In this configuration, the X-ray tube generates a fan-beam X-ray and orbits around the patient. The ring artifact problem identified in Third-generation CT scanners was solved by this configuration. The EMI 7000 series similarly adhered to this principle.&lt;ref&gt;{{Cite book|title=The essential physics of medical imaging|date=2012|publisher=Wolters Kluwer, Lippincott Williams &amp; Wilkins|isbn=978-0-7817-8057-5|editor-last=Bushberg|editor-first=Jerrold T.|edition=3. ed|location=Philadelphia}} p 373.&lt;/ref&gt;

In Fifth generation CT, also know as electron beam computed tomography, both the x ray source and the detectors are stationary. This generation does not use conventional X-ray tube, rather employs a substantial tungsten arc (covering 210°) that surrounds the patient and directly faces the detector ring. An electron gun is utilized to guide and concentrate a rapid electron beam along the tungsten target ring within the gantry.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=SarDDwAAQBAJ&amp;q=ebct&amp;pg=PA6|title=Cardiovascular Computed Tomography|last=Stirrup|first=James|date=2020-01-02|publisher=Oxford University Press|isbn=978-0-19-880927-2|language=en}}&lt;/ref&gt; This type had a major advantage since sweep speeds can be much faster, allowing for less blurry imaging of moving structures, such as the heart and arteries.&lt;ref&gt;{{Cite journal|last1=Talisetti|first1=Anita|last2=Jelnin|first2=Vladimir|last3=Ruiz|first3=Carlos|last4=John|first4=Eunice|last5=Benedetti|first5=Enrico|last6=Testa|first6=Giuliano|last7=Holterman|first7=Ai-Xuan L.|last8=Holterman|first8=Mark J.|date=December 2004|title=Electron beam CT scan is a valuable and safe imaging tool for the pediatric surgical patient|journal=Journal of Pediatric Surgery|volume=39|issue=12|pages=1859–1862|doi=10.1016/j.jpedsurg.2004.08.024|issn=1531-5037|pmid=15616951}}&lt;/ref&gt; Fewer scanners of this design have been produced when compared with spinning tube types, mainly due to the higher cost associated with building a much larger X-ray tube and detector array and limited anatomical coverage.&lt;ref&gt;{{Cite journal|last=Retsky|first=Michael|date=31 July 2008|title=Electron beam computed tomography: Challenges and opportunities|journal=Physics Procedia|volume=1|issue=1|pages=149–154|bibcode=2008PhPro...1..149R|doi=10.1016/j.phpro.2008.07.090|doi-access=free}}&lt;/ref&gt;

=== Classification according to scanning method ===
Sequential CT, also known as step-and-shoot CT, is a scanning method in which the CT table moves stepwise. The process involves the table moving to a specific position, halting for the rotation and acquisition of a slice by the [[w:X-ray tube|X-ray tube]], followed by another incremental movement for the capture of subsequent slices. This method necessitates the table to pause during the slice acquisition, leading to increased scanning time due to interscan delays after each 360° rotation.&lt;ref&gt;{{Cite book |last=Terrier |first=F. |url=https://books.google.com/books?id=AV3wCAAAQBAJ&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA4&amp;dq=Sequential+CT+scan&amp;hl=en |title=Spiral CT of the Abdomen |last2=Grossholz |first2=M. |last3=Becker |first3=C. D. |date=2012-12-06 |publisher=Springer Science &amp; Business Media |isbn=978-3-642-56976-0 |page=4 |language=en}}&lt;/ref&gt; Before the introduction of slip ring technology, this scanning approach was commonly utilized. The need for the tube to return to its initial position after each rotation was essential to prevent cables connecting rotating components, such as the x-ray tube and detectors, from becoming entangled, resulting in prolonged interscan delays.[[File:Drawing of CT fan beam (left) and patient in a CT imaging system.gif|thumb|Drawing of CT fan beam and patient in a CT imaging system|223x223px]]
Spinning tube, commonly called [[w:Spiral computed tomography|spiral CT]], or helical CT, is an imaging technique in which an entire [[w:X-ray tube|X-ray tube]] is spun around the central axis of the area being scanned while the patient table is moving continuously.&lt;ref&gt;{{Cite journal|last=Fuchs|first=Theobald|last2=Kachelrieß|first2=Marc|last3=Kalender|first3=Willi A.|date=2000-11|title=Technical advances in multi–slice spiral CT|url=https://doi.org/10.1016/S0720-048X(00)00269-2|journal=European Journal of Radiology|volume=36|issue=2|pages=69–73|doi=10.1016/s0720-048x(00)00269-2|issn=0720-048X}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last1=Fishman |first1=Elliot K. |url=https://books.google.com/books?id=aWlrAAAAMAAJ&amp;q=spiral+ct |title=Spiral CT: Principles, Techniques, and Clinical Applications |last2=Jeffrey |first2=R. Brooke |date=1995 |publisher=Raven Press |isbn=978-0-7817-0218-8 |language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last=Hsieh |first=Jiang |url=https://books.google.com/books?id=JX__lLLXFHkC&amp;q=spiral+ct&amp;pg=PA265 |title=Computed Tomography: Principles, Design, Artifacts, and Recent Advances |date=2003 |publisher=SPIE Press |isbn=978-0-8194-4425-7 |page=265 |language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Crawford|first=Carl R.|last2=King|first2=Kevin F.|date=1990-11|title=Computed tomography scanning with simultaneous patient translation|url=https://aapm.onlinelibrary.wiley.com/doi/10.1118/1.596464|journal=Medical Physics|language=en|volume=17|issue=6|pages=967–982|doi=10.1118/1.596464|issn=0094-2405}}&lt;/ref&gt; Continuous scanning was made possible by slip ring technology. Slip rings provide an interface through a ring-and-brush arrangement, ensuring uninterrupted electrical connections. This eliminates the requirement for the X-ray tube to return to its initial position after each rotation, enabling continuous movement of the x-ray tube. The scanners with slip ring was introduced in 1987 by Siemens Medical Systems.&lt;ref&gt;{{Cite journal|last=Kalender|first=Willi A.|date=1996|editor-last=Vogl|editor-first=Thomas J.|editor2-last=Clauß|editor2-first=Wolfram|editor3-last=Li|editor3-first=Guo-Zhen|editor4-last=Yeon|editor4-first=Kyung Mo|title=Technical Foundations of Spiral Computed Tomography|url=https://link.springer.com/chapter/10.1007/978-3-642-79887-0_3|journal=Computed Tomography|language=en|location=Berlin, Heidelberg|publisher=Springer|pages=17–28|doi=10.1007/978-3-642-79887-0_3|isbn=978-3-642-79887-0}}&lt;/ref&gt;

=== Classifications according to the X-ray beam geometry ===
Pencil beam computed tomography employs a narrow, parallel X-ray beam geometry, while fan beam CT utilizes an X-ray beam that diverges outward from the radiation source.&lt;ref&gt;{{Cite journal|last=Horiba|first=Isao|last2=Yanaka|first2=Shigenobu|last3=Iwata|first3=Akira|last4=Suzumura|first4=Nobuo|date=1986-01|title=High‐resolution algorithm for fan beam‐CT system|url=https://onlinelibrary.wiley.com/doi/10.1002/scj.4690170703|journal=Systems and Computers in Japan|language=en|volume=17|issue=7|pages=19–29|doi=10.1002/scj.4690170703|issn=0882-1666}}&lt;/ref&gt;&lt;ref name=&quot;auto4&quot;/&gt;

Cone beam computed tomography (CBCT) uses a diverging cone shaped x ray beam for the generation of images. This type is particularly well-suited for dentistry and orthodontics due to its ability to achieve high-resolution imaging with voxel sizes as small as 0.1 mm. Moreover, it offers a considerable advantage over spiral CT by utilizing substantially lower levels of radiation.&lt;ref&gt;{{Cite journal|last=Larson|first=Brent E.|date=2012-04|title=Cone-beam computed tomography is the imaging technique of choice for comprehensive orthodontic assessment|url=https://doi.org/10.1016/j.ajodo.2012.02.009|journal=American Journal of Orthodontics and Dentofacial Orthopedics|volume=141|issue=4|pages=402–410|doi=10.1016/j.ajodo.2012.02.009|issn=0889-5406}}&lt;/ref&gt;

=== Classifications according to the detectors ===
A Single-row CT scanner utilizes a solitary row of detectors, enabling it to gather data for a single slice. Consequently, it is also referred to as Single Slice CT. Multi-row detector CT scanners are equipped with multiple rows of detectors. These detector rows acquire images simultaneously, enabling the rapid acquisition of multiple slices at once. MSCT scanners  offer enhanced image quality but come at the expense of increased radiation exposure compared to their single-slice CT. &lt;ref&gt;{{Cite journal|date=2008-08-20|title=Annals of the ICRP, Publication 102, Managing Patient Dose in Multi-Detector Computed Tomography (MDCT) Radiation Dose from Adult and Pediatric Multidetector Computed Tomography COMARE 12th Report, The Impact of Personally Initiated X-ray Computed Tomography Scanning for the Health Assessment of Asymptomatic Individuals The Psychology of Risk|url=http://dx.doi.org/10.1088/0952-4746/28/3/b01|journal=Journal of Radiological Protection|volume=28|issue=3|pages=435–441|doi=10.1088/0952-4746/28/3/b01|issn=0952-4746}}&lt;/ref&gt;

Photon-counting computed tomography  is a recent advancement in computed tomography that employs a photon-counting detector  to detect X-rays, registering the interactions of individual photons. Through monitoring the deposited energy in each interaction, the detectors capture an approximate energy spectrum.&lt;ref&gt;{{Cite journal|last=Willemink|first=Martin J.|last2=Persson|first2=Mats|last3=Pourmorteza|first3=Amir|last4=Pelc|first4=Norbert J.|last5=Fleischmann|first5=Dominik|date=2018-11|title=Photon-counting CT: Technical Principles and Clinical Prospects|url=http://pubs.rsna.org/doi/10.1148/radiol.2018172656|journal=Radiology|language=en|volume=289|issue=2|pages=293–312|doi=10.1148/radiol.2018172656|issn=0033-8419}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Leng|first=Shuai|last2=Bruesewitz|first2=Michael|last3=Tao|first3=Shengzhen|last4=Rajendran|first4=Kishore|last5=Halaweish|first5=Ahmed F.|last6=Campeau|first6=Norbert G.|last7=Fletcher|first7=Joel G.|last8=McCollough|first8=Cynthia H.|date=2019-05|title=Photon-counting Detector CT: System Design and Clinical Applications of an Emerging Technology|url=http://pubs.rsna.org/doi/10.1148/rg.2019180115|journal=RadioGraphics|language=en|volume=39|issue=3|pages=729–743|doi=10.1148/rg.2019180115|issn=0271-5333|pmc=PMC6542627|pmid=31059394}}&lt;/ref&gt;

Flat Panel CT represents a CT scanner which is characterized by the integration of flat panel detectors. These scanners present a significant advancement in volumetric coverage, facilitating comprehensive imaging of entire organs such as the heart, kidneys, or brain through a singular axial scan. &lt;ref&gt;{{Cite journal|last=Grasruck|first=M.|last2=Suess|first2=Ch|last3=Stierstorfer|first3=K.|last4=Popescu|first4=S.|last5=Flohr|first5=T.|date=2005-04-20|title=Evaluation of image quality and dose on a flat-panel CT-scanner|url=https://www.spiedigitallibrary.org/conference-proceedings-of-spie/5745/0000/Evaluation-of-image-quality-and-dose-on-a-flat-panel/10.1117/12.594583.full|journal=Medical Imaging 2005: Physics of Medical Imaging|publisher=SPIE|volume=5745|pages=179–188|doi=10.1117/12.594583}}&lt;/ref&gt;&lt;ref name=&quot;auto5&quot;&gt;{{Cite journal|last=Gupta|first=Rajiv|last2=Cheung|first2=Arnold C.|last3=Bartling|first3=Soenke H.|last4=Lisauskas|first4=Jennifer|last5=Grasruck|first5=Michael|last6=Leidecker|first6=Christianne|last7=Schmidt|first7=Bernhard|last8=Flohr|first8=Thomas|last9=Brady|first9=Thomas J.|date=2008-11|title=Flat-Panel Volume CT: Fundamental Principles, Technology, and Applications|url=http://pubs.rsna.org/doi/10.1148/rg.287085004|journal=RadioGraphics|language=en|volume=28|issue=7|pages=2009–2022|doi=10.1148/rg.287085004|issn=0271-5333}}&lt;/ref&gt;&lt;ref name=&quot;auto5&quot;/&gt;

=== Dual Energy CT ===
Dual Energy CT also known as Spectral CT is an advancement of Computed Tomography in which two energies are used to create two sets of data.&lt;ref&gt;{{Cite book |last=Johnson |first=Thorsten |url=https://books.google.com/books?id=Etvcnz0mjF4C&amp;newbks=0&amp;printsec=frontcover&amp;dq=dual+energy+ct&amp;hl=en |title=Dual Energy CT in Clinical Practice |last2=Fink |first2=Christian |last3=Schönberg |first3=Stefan O. |last4=Reiser |first4=Maximilian F. |date=2011-01-18 |publisher=Springer Science &amp; Business Media |isbn=978-3-642-01740-7 |language=en}}&lt;/ref&gt; A Dual Energy CT may employ Dual source, Single source with dual detector layer, Single source with energy switching methods to get two different sets of data.&lt;ref&gt;{{Cite book |last=Johnson |first=Thorsten |url=https://books.google.com/books?id=Etvcnz0mjF4C&amp;newbks=0&amp;printsec=frontcover&amp;dq=dual+energy+ct&amp;hl=en |title=Dual Energy CT in Clinical Practice |last2=Fink |first2=Christian |last3=Schönberg |first3=Stefan O. |last4=Reiser |first4=Maximilian F. |date=2011-01-18 |publisher=Springer Science &amp; Business Media |isbn=978-3-642-01740-7 |page=8 |language=en}}&lt;/ref&gt; It was commercially introduced in 2006.&lt;ref&gt;{{Cite journal|last=Schmidt|first=Bernhard|last2=Flohr|first2=Thomas|date=2020-11|title=Principles and applications of dual source CT|url=https://linkinghub.elsevier.com/retrieve/pii/S112017972030257X|journal=Physica Medica|language=en|volume=79|pages=36–46|doi=10.1016/j.ejmp.2020.10.014}}&lt;/ref&gt; Dual source CT is an advanced scanner with a two X-ray tube detector system, unlike conventional single tube systems.&lt;ref&gt;{{Cite book |last1=Carrascosa |first1=Patricia M. |url=https://books.google.com/books?id=wJ2oCgAAQBAJ&amp;q=dual+source+ct |title=Dual-Energy CT in Cardiovascular Imaging |last2=Cury |first2=Ricardo C. |last3=García |first3=Mario J. |last4=Leipsic |first4=Jonathon A. |date=2015-10-03 |publisher=Springer |isbn=978-3-319-21227-2 |language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal |last1=Schmidt |first1=Bernhard |last2=Flohr |first2=Thomas |date=2020-11-01 |title=Principles and applications of dual source CT |url=https://www.sciencedirect.com/science/article/pii/S112017972030257X |journal=Physica Medica |series=125 Years of X-Rays |language=en |volume=79 |pages=36–46 |doi=10.1016/j.ejmp.2020.10.014 |issn=1120-1797 |pmid=33115699 |s2cid=226056088}}&lt;/ref&gt; These two detector systems are mounted on a single gantry at 90° in the same plane.&lt;ref name=&quot;auto1&quot;&gt;{{Cite book |last1=Seidensticker |first1=Peter R. |url=https://books.google.com/books?id=oUtHea3ZnJ0C&amp;q=dual+source+ct |title=Dual Source CT Imaging |last2=Hofmann |first2=Lars K. |date=2008-05-24 |publisher=Springer Science &amp; Business Media |isbn=978-3-540-77602-4 |language=en}}&lt;/ref&gt;  This scanner allow fast scanning with higher temporal resolution by acquiring a full CT slice in only half a rotation. Fast imaging reduces motion blurring at high heart rates and potentially allowing for shorter breath-hold time. This is particularly useful for ill patients having difficulty holding their breath or unable to take heart-rate lowering medication.&lt;ref name=&quot;auto1&quot; /&gt;&lt;ref&gt;{{Cite journal |last1=Schmidt |first1=Bernhard |last2=Flohr |first2=Thomas |date=2020-11-01 |title=Principles and applications of dual source CT |url=https://www.physicamedica.com/article/S1120-1797(20)30257-X/abstract |journal=Physica Medica: European Journal of Medical Physics |language=English |volume=79 |pages=36–46 |doi=10.1016/j.ejmp.2020.10.014 |issn=1120-1797 |pmid=33115699 |s2cid=226056088}}&lt;/ref&gt;

Single Source with Energy switching is another mode of Dual energy CT in which a single tube is operated at two different energies by switching the energies frequently.&lt;ref&gt;{{Cite journal |last=Mahmood |first=Usman |last2=Horvat |first2=Natally |last3=Horvat |first3=Joao Vicente |last4=Ryan |first4=Davinia |last5=Gao |first5=Yiming |last6=Carollo |first6=Gabriella |last7=DeOcampo |first7=Rommel |last8=Do |first8=Richard K. |last9=Katz |first9=Seth |last10=Gerst |first10=Scott |last11=Schmidtlein |first11=C. Ross |last12=Dauer |first12=Lawrence |last13=Erdi |first13=Yusuf |last14=Mannelli |first14=Lorenzo |date=May 2018 |title=Rapid Switching kVp Dual Energy CT: Value of Reconstructed Dual Energy CT Images and Organ Dose Assessment in Multiphasic Liver CT Exams |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5918634/ |journal=European journal of radiology |volume=102 |pages=102–108 |doi=10.1016/j.ejrad.2018.02.022 |issn=0720-048X |pmc=5918634 |pmid=29685522}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal |last=Johnson |first=Thorsten R. C. |date=November 2012 |title=Dual-Energy CT: General Principles |url=https://www.ajronline.org/doi/10.2214/AJR.12.9116 |journal=American Journal of Roentgenology |language=en |volume=199 |issue=5_supplement |pages=S3–S8 |doi=10.2214/AJR.12.9116 |issn=0361-803X}}&lt;/ref&gt; Dual layer spectral CT is a sub-type in which the spectral data is obtained by using two separate scintillator layers. It consists of two detector layer in which one is on the top of another. The detector layer that is closer to the x ray tube detects the low energy x rays and lets the high energy x rays to pass to the layer that is below. The high energy x rays are detected by the second layer.&lt;ref&gt;{{Cite journal|last=Ehn|first=Sebastian|last2=Sellerer|first2=Thorsten|last3=Muenzel|first3=Daniela|last4=Fingerle|first4=Alexander A.|last5=Kopp|first5=Felix|last6=Duda|first6=Manuela|last7=Mei|first7=Kai|last8=Renger|first8=Bernhard|last9=Herzen|first9=Julia|date=2018-01|title=Assessment of quantification accuracy and image quality of a full‐body dual‐layer spectral CT system|url=https://onlinelibrary.wiley.com/doi/10.1002/acm2.12243|journal=Journal of Applied Clinical Medical Physics|language=en|volume=19|issue=1|pages=204–217|doi=10.1002/acm2.12243|issn=1526-9914|pmc=PMC5768037|pmid=29266724}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Rassouli|first=Negin|last2=Etesami|first2=Maryam|last3=Dhanantwari|first3=Amar|last4=Rajiah|first4=Prabhakar|date=2017-12-01|title=Detector-based spectral CT with a novel dual-layer technology: principles and applications|url=https://doi.org/10.1007/s13244-017-0571-4|journal=Insights into Imaging|language=en|volume=8|issue=6|pages=589–598|doi=10.1007/s13244-017-0571-4|issn=1869-4101|pmc=PMC5707218|pmid=28986761}}&lt;/ref&gt;

=== Hybrid CT imaging ===
Hybrid imaging involves integrating two or more imaging modalities to create a novel technique. This fusion leverages the inherent strengths of the combined imaging technologies, resulting in the emergence of a more potent and advanced modality.[[File:Petct1.jpg|thumb|PET-CT scan of chest|161x161px]]
Positron emission tomography–computed tomography is a hybrid CT modality which combines, in a single gantry, a [[w:positron emission tomography|positron emission tomography]] (PET) scanner and an x-ray computed tomography scanner, to acquire sequential images from both devices in the same session, which are combined into a single superposed ([[w:Image registration|co-registered]]) image. Thus, [[w:functional imaging|functional imaging]] obtained by PET, which depicts the spatial distribution of metabolic or biochemical activity in the body can be more precisely aligned or correlated with anatomic imaging obtained by CT scanning.&lt;ref&gt;{{Cite journal |last=Blodgett |first=Todd M. |last2=Meltzer |first2=Carolyn C. |last3=Townsend |first3=David W. |date=February 2007 |title=PET/CT: form and function |url=https://pubmed.ncbi.nlm.nih.gov/17255408/#:~:text=CT%20is%20complementary%20in%20the,identify%20and%20localize%20functional%20abnormalities. |journal=Radiology |volume=242 |issue=2 |pages=360–385 |doi=10.1148/radiol.2422051113 |issn=0033-8419 |pmid=17255408}}&lt;/ref&gt; PET-CT gives both anatomical and functional details of an organ under examination and is helpful in detecting different type of cancers.&lt;ref&gt;{{Cite journal |last=Ciernik |first=I.Frank |last2=Dizendorf |first2=Elena |last3=Baumert |first3=Brigitta G |last4=Reiner |first4=Beatrice |last5=Burger |first5=Cyrill |last6=Davis |first6=J.Bernard |last7=Lütolf |first7=Urs M |last8=Steinert |first8=Hans C |last9=Von Schulthess |first9=Gustav K |date=November 2003 |title=Radiation treatment planning with an integrated positron emission and computer tomography (PET/CT): a feasibility study |url=https://doi.org/10.1016/S0360-3016(03)00346-8 |journal=International Journal of Radiation Oncology*Biology*Physics |volume=57 |issue=3 |pages=853–863 |doi=10.1016/s0360-3016(03)00346-8 |issn=0360-3016}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal |last=Ul-Hassan |first=Fahim |last2=Cook |first2=Gary J |date=August 2012 |title=PET/CT in oncology |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4952129/ |journal=Clinical Medicine |volume=12 |issue=4 |pages=368–372 |doi=10.7861/clinmedicine.12-4-368 |issn=1470-2118 |pmc=4952129 |pmid=22930885}}&lt;/ref&gt; Hybrid PET-CT systems have become more effective with the integration of anatomical details from CT scans. This integration allows for the creation of an attenuation correction map, which helps refine PET images. These advancements have notably reduced examination duration, increased diagnostic accuracy, and instilled greater confidence in the accuracy of diagnoses.&lt;ref&gt;{{Cite journal|last=Veit-Haibach|first=Patrick|last2=Luczak|first2=Christopher|last3=Wanke|first3=Isabel|last4=Fischer|first4=Markus|last5=Egelhof|first5=Thomas|last6=Beyer|first6=Thomas|last7=Dahmen|first7=Gerlinde|last8=Bockisch|first8=Andreas|last9=Rosenbaum|first9=Sandra|date=2007-12-01|title=TNM staging with FDG-PET/CT in patients with primary head and neck cancer|url=https://doi.org/10.1007/s00259-007-0564-5|journal=European Journal of Nuclear Medicine and Molecular Imaging|language=en|volume=34|issue=12|pages=1953–1962|doi=10.1007/s00259-007-0564-5|issn=1619-7089}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Sonni|first=Ida|last2=Baratto|first2=Lucia|last3=Park|first3=Sonya|last4=Hatami|first4=Negin|last5=Srinivas|first5=Shyam|last6=Davidzon|first6=Guido|last7=Gambhir|first7=Sanjiv Sam|last8=Iagaru|first8=Andrei|date=2018-04-18|title=Initial experience with a SiPM-based PET/CT scanner: influence of acquisition time on image quality|url=https://doi.org/10.1186/s40658-018-0207-x|journal=EJNMMI Physics|volume=5|issue=1|pages=9|doi=10.1186/s40658-018-0207-x|issn=2197-7364|pmc=PMC5904089|pmid=29666972}}&lt;/ref&gt; In oncology, studies show that using PET-CT together is better for accurately staging and restaging than using CT or PET alone.&lt;ref&gt;{{Cite journal|last=Ben-Haim|first=Simona|last2=Ell|first2=Peter|date=2009-01-01|title=18F-FDG PET and PET/CT in the Evaluation of Cancer Treatment Response|url=https://jnm.snmjournals.org/content/50/1/88|journal=Journal of Nuclear Medicine|language=en|volume=50|issue=1|pages=88–99|doi=10.2967/jnumed.108.054205|issn=0161-5505|pmid=19139187}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Czernin|first=Johannes|last2=Allen-Auerbach|first2=Martin|last3=Schelbert|first3=Heinrich R.|date=2007-01-01|title=Improvements in Cancer Staging with PET/CT: Literature-Based Evidence as of September 2006|url=https://jnm.snmjournals.org/content/48/1_suppl/78S|journal=Journal of Nuclear Medicine|language=en|volume=48|issue=1 suppl|pages=78S–88S|issn=0161-5505|pmid=17204723}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Ul-Hassan|first=Fahim|last2=Cook|first2=Gary J|date=2012-08|title=PET/CT in oncology|url=https://www.rcpjournals.org/lookup/doi/10.7861/clinmedicine.12-4-368|journal=Clinical Medicine|language=en|volume=12|issue=4|pages=368–372|doi=10.7861/clinmedicine.12-4-368|issn=1470-2118|pmc=PMC4952129|pmid=22930885}}&lt;/ref&gt;

Single photon emission computed tomography- computed tomography also known as SPECT-CT is a hybrid imaging modality, used in Nuclear Medicine which combines a [[w:Single-photon emission computed tomography|SPECT]] scanner and a CT scanner into one machine. This hybrid modality was first introduced commercially in 1999&lt;ref name=&quot;auto6&quot;&gt;{{Cite journal|last=Ritt|first=P.|last2=Sanders|first2=J.|last3=Kuwert|first3=T.|date=2014-12-01|title=SPECT/CT technology|url=https://doi.org/10.1007/s40336-014-0086-7|journal=Clinical and Translational Imaging|language=en|volume=2|issue=6|pages=445–457|doi=10.1007/s40336-014-0086-7|issn=2281-7565}}&lt;/ref&gt; SPECT-CT uses a radiotracer for evaluation of function details and x rays anatomical details. These image sets are then coregistered to allow for a comprehensive understanding of the relationship between physiological function and anatomical structures, aiding in more accurate and reliable diagnostic evaluations.&lt;ref name=&quot;auto6&quot;/&gt;

Angio-CT is a hybrid machine which combines the fluoroscopic angiographic imaging and cross-sectional imaging of CT. These systems offer an integrated approach that combines the benefits of conventional angiography with the imaging capabilities of CT technology.&lt;ref name=&quot;auto2&quot;&gt;{{Cite journal|last=Taiji|first=Ryosuke|last2=Lin|first2=Ethan Y.|last3=Lin|first3=Yuan-Mao|last4=Yevich|first4=Steven|last5=Avritscher|first5=Rony|last6=Sheth|first6=Rahul A.|last7=Ruiz|first7=Joseph R.|last8=Jones|first8=A. Kyle|last9=Chintalapani|first9=Gouthami|date=2021-09-01|title=Combined Angio-CT Systems: A Roadmap Tool for Precision Therapy in Interventional Oncology|url=http://pubs.rsna.org/doi/10.1148/rycan.2021210039|journal=Radiology: Imaging Cancer|language=en|volume=3|issue=5|pages=e210039|doi=10.1148/rycan.2021210039|issn=2638-616X|pmc=PMC8489448|pmid=34559007}}&lt;/ref&gt; This hybrid imaging modality was introduced by Yasuaki Arai in 1992.&lt;ref&gt;{{Cite journal|last=Yoshitaka|first=Inaba|last2=Yasuaki|first2=Arai|last3=Yoshito|first3=Takeuchi|last4=Hideyuki|first4=Takeda|last5=Toyohiro|first5=Ota|last6=Satoru|first6=Sueyoshi|last7=Takuji|first7=Yamagami|last8=Kazuhiko|first8=Ohashi|last9=K|first9=Yun|date=1996|title=New Diagnostic Imagings for IVR. Clinical Effectiveness of a Newly Developed Interventional-CT system.|url=https://jglobal.jst.go.jp/en/detail?JGLOBAL_ID=200902150551428412|journal=IVR|language=en|volume=11|issue=1|pages=43–49|issn=1340-4520}}&lt;/ref&gt;&lt;ref name=&quot;auto2&quot;/&gt;

== Medical use ==
Since its introduction in the 1970s,&lt;ref&gt;{{Cite book |last1=Curry |first1=Thomas S. |url=https://books.google.com/books?id=W2PrMwHqXl0C |title=Christensen's Physics of Diagnostic Radiology |last2=Dowdey |first2=James E. |last3=Murry |first3=Robert C. |date=1990 |publisher=Lippincott Williams &amp; Wilkins |isbn=978-0-8121-1310-5 |pages=289 |language=en}}&lt;/ref&gt; CT has become an important tool in [[w:medical imaging|medical imaging]] to supplement conventional [[w:X-ray|X-ray]] imaging and [[w:medical ultrasonography|medical ultrasonography]]. It has more recently been used for [[w:preventive medicine|preventive medicine]] or [[w:screening (medicine)|screening]] for disease, for example, [[w:Virtual colonoscopy|CT colonography]] for people with a high risk of [[w:colon cancer|colon cancer]], or full-motion heart scans for people with a high risk of heart disease. The use of CT scans has increased dramatically over the last two decades in many countries.&lt;ref name=&quot;Smith2009&quot;&gt;{{Cite journal |vauthors=Smith-Bindman R, Lipson J, Marcus R, Kim KP, Mahesh M, Gould R, Berrington de González A, [[Diana Miglioretti|Miglioretti DL]] |date=December 2009 |title=Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer |journal=Arch. Intern. Med. |volume=169 |issue=22 |pages=2078–86 |doi=10.1001/archinternmed.2009.427 |pmc=4635397 |pmid=20008690}}&lt;/ref&gt;  An estimated 72 million scans were performed in the United States in 2007 and more than 80 million in 2015.&lt;ref name=&quot;Berrington2009&quot;&gt;{{Cite journal |vauthors=Berrington de González A, Mahesh M, Kim KP, Bhargavan M, Lewis R, Mettler F, Land C |date=December 2009 |title=Projected cancer risks from computed tomographic scans performed in the United States in 2007 |journal=Arch. Intern. Med. |volume=169 |issue=22 |pages=2071–7 |doi=10.1001/archinternmed.2009.440 |pmc=6276814 |pmid=20008689}}&lt;/ref&gt;&lt;ref&gt;{{Cite web |title=Dangers of CT Scans and X-Rays – Consumer Reports |url=https://www.consumerreports.org/cro/magazine/2015/01/the-surprising-dangers-of-ct-sans-and-x-rays/index.htm |access-date=16 May 2018}}&lt;/ref&gt;

=== Diagnostic ===

==== Head &amp; Neck Imaging ====
[[File:CT of a normal brain (thumbnail).png|thumb|218x218px|CT Head of a normal brain: Sagittal (top), Coronal (bottom left), Axial (bottom right)]]
CT scan remains the cornerstone imaging modality for the initial evaluation and subsequent management of patients with acute traumatic brain injury due to its rapid acquisition time and high sensitivity for detecting hemorrhagic complications, such as intraparenchymal hematomas and subdural hemorrhages.&lt;ref&gt;{{Cite journal|last=Schweitzer|first=Andrew D.|last2=Niogi|first2=Sumit N.|last3=Whitlow|first3=Christopher J.|last4=Tsiouris|first4=A. John|date=2019-10|title=Traumatic Brain Injury: Imaging Patterns and Complications|url=http://pubs.rsna.org/doi/10.1148/rg.2019190076|journal=RadioGraphics|language=en|volume=39|issue=6|pages=1571–1595|doi=10.1148/rg.2019190076|issn=0271-5333}}&lt;/ref&gt; CT scan of the head is typically used to detect [[w:infarction|infarction]] ([[w:stroke|stroke]]), [[w:Neoplasm|tumors]], [[w:calcification|calcification]]s, [[w:haemorrhage|haemorrhage]].&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=pUVwDwAAQBAJ&amp;q=CT+scanning+of+the+head+is+typically+used+to+detect&amp;pg=PA389|title=Critical Care Transport|last1=Surgeons (AAOS)|first1=American Academy of Orthopaedic|last2=Physicians (ACEP)|first2=American College of Emergency|last3=UMBC|date=2017-03-20|publisher=Jones &amp; Bartlett Learning|isbn=978-1-284-04099-9|page=389|language=en}}&lt;/ref&gt; Tumors can be detected by the swelling and anatomical distortion they cause, or by surrounding edema. CT scanning of the head is also used in CT-[[w:image guided surgery|guided]] [[w:stereotactic surgery|stereotactic surgery]] and [[w:radiosurgery|radiosurgery]] for treatment of intracranial tumors, [[w:arteriovenous malformation|arteriovenous malformation]]s, and other surgically treatable conditions using a device known as the [[w:N-localizer|N-localizer]].&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=ioxongEACAAJ|title=Image-Guided Neurosurgery|last=Galloway|first=RL Jr.|publisher=Elsevier|year=2015|isbn=978-0-12-800870-6|editor-last=Golby|editor-first=AJ|location=Amsterdam|pages=3–4|chapter=Introduction and Historical Perspectives on Image-Guided Surgery}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=uEghr21XY6wC|title=Principles and Practice of Stereotactic Radiosurgery|last1=Tse|first1=VCK|last2=Kalani|first2=MYS|last3=Adler|first3=JR|publisher=Springer|year=2015|isbn=978-0-387-71070-9|editor-last=Chin|editor-first=LS|location=New York|page=28|chapter=Techniques of Stereotactic Localization|editor-last2=Regine|editor-first2=WF}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|title=Stereotactic Radiosurgery and Stereotactic Body Radiation Therapy|last1=Saleh|first1=H|last2=Kassas|first2=B|publisher=CRC Press|year=2015|isbn=978-1-4398-4198-3|editor-last=Benedict|editor-first=SH|location=Boca Raton|pages=156–159|chapter=Developing Stereotactic Frames for Cranial Treatment|editor-last2=Schlesinger|editor-first2=DJ|editor-last3=Goetsch|editor-first3=SJ|editor-last4=Kavanagh|editor-first4=BD|chapter-url=https://books.google.com/books?id=Pm3RBQAAQBAJ&amp;q=Developing+Stereotactic+Frames+for+Cranial+Treatment&amp;pg=PA153}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=mAN3MAEACAAJ&amp;q=0444534970|title=Brain Stimulation|last1=Khan|first1=FR|last2=Henderson|first2=JM|journal=Handbook of Clinical Neurology|publisher=Elsevier|year=2013|isbn=978-0-444-53497-2|editor-last=Lozano|editor-first=AM|volume=116|location=Amsterdam|pages=28–30|chapter=Deep Brain Stimulation Surgical Techniques|doi=10.1016/B978-0-444-53497-2.00003-6|pmid=24112882|editor-last2=Hallet|editor-first2=M}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=cnF-2KCeR1sC&amp;q=Textbook+of+Stereotactic+and+Functional+Neurosurgery|title=Textbook of Stereotactic and Functional Neurosurgery|last=Arle|first=J|publisher=Springer-Verlag|year=2009|isbn=978-3-540-69959-0|editor-last=Lozano|editor-first=AM|location=Berlin|pages=456–461|chapter=Development of a Classic: the Todd-Wells Apparatus, the BRW, and the CRW Stereotactic Frames|editor-last2=Gildenberg|editor-first2=PL|editor-last3=Tasker|editor-first3=RR}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=June 2012|title=Invention of the N-localizer for stereotactic neurosurgery and its use in the Brown-Roberts-Wells stereotactic frame|journal=Neurosurgery|volume=70|issue=2 Supplement Operative|pages=173–176|doi=10.1227/NEU.0b013e318246a4f7|pmid=22186842|vauthors=Brown RA, Nelson JA|s2cid=36350612}}&lt;/ref&gt;

[[w:Contrast CT|Contrast CT]] is generally the initial study of choice for [[w:neck mass|neck mass]]es in adults.&lt;ref&gt;{{Cite journal|last=Das|first=Riya|last2=Sarkar|first2=Tanmoy|last3=Verma|first3=Sweta|date=2022-12|title=A Case Series on Unusual Neck Masses|url=https://link.springer.com/10.1007/s12070-021-02866-5|journal=Indian Journal of Otolaryngology and Head &amp; Neck Surgery|language=en|volume=74|issue=S3|pages=5531–5535|doi=10.1007/s12070-021-02866-5|issn=2231-3796|pmc=PMC8461437|pmid=34584897}}&lt;/ref&gt; [[w:Computed tomography of the thyroid|CT of the thyroid]] plays an important role in the evaluation of [[w:thyroid cancer|thyroid cancer]].&lt;ref name=&quot;Saeedan2016&quot;&gt;{{Cite journal|last1=Bin Saeedan|first1=Mnahi|last2=Aljohani|first2=Ibtisam Musallam|last3=Khushaim|first3=Ayman Omar|last4=Bukhari|first4=Salwa Qasim|last5=Elnaas|first5=Salahudin Tayeb|year=2016|title=Thyroid computed tomography imaging: pictorial review of variable pathologies|journal=Insights into Imaging|volume=7|issue=4|pages=601–617|doi=10.1007/s13244-016-0506-5|issn=1869-4101|pmc=4956631|pmid=27271508}}&lt;/ref&gt; CT scan often incidentally finds thyroid abnormalities, and so is often the preferred investigation modality for thyroid abnormalities.&lt;ref name=&quot;Saeedan2016&quot; /&gt;

==== Body Imaging ====
A CT scan can be used for detecting both acute and chronic changes in the [[w:Parenchyma#Lung parenchyma|lung parenchyma]], the tissue of the [[w:lung|lung]]s.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=rQlDDwAAQBAJ|title=Computed Tomography of the Lung|publisher=Springer Berlin Heidelberg|year=2007|isbn=978-3-642-39518-5|pages=40, 47}}&lt;/ref&gt; It is particularly relevant here because normal two-dimensional X-rays do not show such defects. A variety of techniques are used, depending on the suspected abnormality. For evaluation of chronic interstitial processes such as [[w:Pneumatosis#Lungs|emphysema]], and [[w:Pulmonary fibrosis#|fibrosis]],&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=VKATAQAAMAAJ&amp;q=ct+of+lungs|title=High-resolution CT of the Lung|publisher=Lippincott Williams &amp; Wilkins|year=2009|isbn=978-0-7817-6909-9|pages=81,568}}&lt;/ref&gt; thin sections with high spatial frequency reconstructions are used; often scans are performed both on inspiration and expiration. This special technique is called [[w:high resolution CT|high resolution CT]] that produces a sampling of the lung, and not continuous images.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=QjouDwAAQBAJ&amp;q=HRCT|title=Specialty Imaging: HRCT of the Lung E-Book|last1=Martínez-Jiménez|first1=Santiago|last2=Rosado-de-Christenson|first2=Melissa L.|last3=Carter|first3=Brett W.|date=2017-07-22|publisher=Elsevier Health Sciences|isbn=978-0-323-52495-7|language=en}}&lt;/ref&gt;

CT is an accurate technique for diagnosis of [[w:Human abdomen|abdominal]] diseases like [[w:Crohn's disease|Crohn's disease]],&lt;ref&gt;{{Cite journal|last1=Furukawa|first1=Akira|last2=Saotome|first2=Takao|last3=Yamasaki|first3=Michio|last4=Maeda|first4=Kiyosumi|last5=Nitta|first5=Norihisa|last6=Takahashi|first6=Masashi|last7=Tsujikawa|first7=Tomoyuki|last8=Fujiyama|first8=Yoshihide|last9=Murata|first9=Kiyoshi|date=2004-05-01|title=Cross-sectional Imaging in Crohn Disease|journal=RadioGraphics|volume=24|issue=3|pages=689–702|doi=10.1148/rg.243035120|issn=0271-5333|pmid=15143222|doi-access=free|last10=Sakamoto|first10=Tsutomu}}&lt;/ref&gt; GIT bleeding, and diagnosis and staging of cancer, as well as follow-up after cancer treatment to assess response.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=r3uK7sSZUmcC|title=CT of the Acute Abdomen|publisher=Springer Berlin Heidelberg|year=2011|isbn=978-3-540-89232-8|pages=37}}&lt;/ref&gt; It is commonly used to investigate [[w:acute abdominal pain|acute abdominal pain]].&lt;ref&gt;{{Cite book|title=Diseases of the Abdomen and Pelvis|last1=Jay P Heiken|last2=Douglas S Katz|publisher=Springer Milan|year=2014|isbn=9788847056596|editor-last=J. Hodler|page=3|chapter=Emergency Radiology of the Abdomen and Pelvis: Imaging of the Nontraumatic and Traumatic Acute Abdomen|editor-last2=R. A. Kubik-Huch|editor-last3=G. K. von Schulthess|editor-last4=Ch. L. Zollikofer|chapter-url=https://books.google.com/books?id=CSy5BQAAQBAJ&amp;pg=PA3}}&lt;/ref&gt; Non-enhanced computed tomography is today the gold standard for diagnosing [[w:Kidney stone disease|urinary stones]].&lt;ref&gt;{{Cite book|url=https://uroweb.org/guidelines/urolithiasis|title=EAU Guidelines on Urolithiasis|last1=Skolarikos|first1=A|last2=Neisius|first2=A|last3=Petřík|first3=A|last4=Somani|first4=B|last5=Thomas|first5=K|last6=Gambaro|first6=G|date=March 2022|publisher=[[European Association of Urology]]|isbn=978-94-92671-16-5|location=Amsterdam}}&lt;/ref&gt; The size, volume and density of stones can be estimated to help clinicians guide further treatment; size is especially important in predicting spontaneous passage of a stone.&lt;ref&gt;{{Cite journal|last1=Miller|first1=Oren F.|last2=Kane|first2=Christopher J.|date=September 1999|title=Time to stone passage for observed ureteral calculi: a guide for patient education|journal=Journal of Urology|volume=162|issue=3 Part 1|pages=688–691|doi=10.1097/00005392-199909010-00014|pmid=10458343}}&lt;/ref&gt;

==== Musculoskeletal Imaging ====
CT scan is widely used for imaging of muscluloskeltal. For the [[w:axial skeleton|axial skeleton]] and [[w:Limb (anatomy)|extremities]], CT is often used to image complex [[w:fracture (bone)|fractures]], especially ones around joints, because of its ability to reconstruct the area of interest in multiple planes.&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/174_2017_25|title=Clinical Application of Musculoskeletal CT: Trauma, Oncology, and Postsurgery|last=Gondim Teixeira|first=Pedro Augusto|last2=Blum|first2=Alain|date=2019|publisher=Springer International Publishing|isbn=978-3-319-42586-3|editor-last=Nikolaou|editor-first=Konstantin|series=Medical Radiology|location=Cham|pages=1079–1105|language=en|doi=10.1007/174_2017_25|editor-last2=Bamberg|editor-first2=Fabian|editor-last3=Laghi|editor-first3=Andrea|editor-last4=Rubin|editor-first4=Geoffrey D.}}&lt;/ref&gt; Fractures, ligamentous injuries, and [[w:Dislocation (medicine)|dislocations]] can easily be recognized with a 0.2&amp;nbsp;mm resolution.&lt;ref&gt;{{Cite web|url=http://orthoinfo.aaos.org/topic.cfm?topic=A00391|title=Ankle Fractures|website=orthoinfo.aaos.org|publisher=American Association of Orthopedic Surgeons|archive-url=https://web.archive.org/web/20100530103553/http://orthoinfo.aaos.org/topic.cfm?topic=A00391|archive-date=30 May 2010|access-date=30 May 2010|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Buckwalter, Kenneth A.|display-authors=etal|date=11 September 2000|title=Musculoskeletal Imaging with Multislice CT|journal=American Journal of Roentgenology|volume=176|issue=4|pages=979–986|doi=10.2214/ajr.176.4.1760979|pmid=11264094}}&lt;/ref&gt; With modern dual-energy CT scanners, new areas of use have been established, such as aiding in the diagnosis of [[w:gout|gout]].&lt;ref&gt;{{Cite journal|last1=Ramon|first1=André|last2=Bohm-Sigrand|first2=Amélie|last3=Pottecher|first3=Pierre|last4=Richette|first4=Pascal|last5=Maillefert|first5=Jean-Francis|last6=Devilliers|first6=Herve|last7=Ornetti|first7=Paul|date=2018-03-01|title=Role of dual-energy CT in the diagnosis and follow-up of gout: systematic analysis of the literature|journal=Clinical Rheumatology|volume=37|issue=3|pages=587–595|doi=10.1007/s10067-017-3976-z|issn=0770-3198|pmid=29350330|s2cid=3686099}}&lt;/ref&gt;

==== Perfusion Imaging ====
[[File:CT perfusion in M1 artery occlusion.png|thumb|CT Perfusion images of Brain with Time to Peak, Cerebral blood volume.|195x195px]]CT perfusion imaging is a specific form of CT to assess flow through [[w:blood vessel|blood vessel]]s whilst injecting a [[w:contrast agent|contrast agent]].&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last1=Wittsack|first1=H.-J.|last2=Wohlschläger|first2=A.M.|last3=Ritzl|first3=E.K.|last4=Kleiser|first4=R.|last5=Cohnen|first5=M.|last6=Seitz|first6=R.J.|last7=Mödder|first7=U.|date=2008-01-01|title=CT-perfusion imaging of the human brain: Advanced deconvolution analysis using circulant singular value decomposition|journal=Computerized Medical Imaging and Graphics|language=en|volume=32|issue=1|pages=67–77|doi=10.1016/j.compmedimag.2007.09.004|issn=0895-6111|pmid=18029143}}&lt;/ref&gt; Blood flow, blood transit time, and organ blood volume, can all be calculated with reasonable [[w:sensitivity and specificity|sensitivity and specificity]].&lt;ref name=&quot;:0&quot; /&gt; This type of CT may be used on the [[w:heart|heart]], although sensitivity and specificity for detecting abnormalities are still lower than for other forms of CT.&lt;ref&gt;{{Cite journal|last1=Williams|first1=M.C.|last2=Newby|first2=D.E.|date=2016-08-01|title=CT myocardial perfusion imaging: current status and future directions|journal=Clinical Radiology|language=en|volume=71|issue=8|pages=739–749|doi=10.1016/j.crad.2016.03.006|issn=0009-9260|pmid=27091433}}&lt;/ref&gt; This may also be used on the [[w:brain|brain]], where CT perfusion imaging can often detect poor brain perfusion well before it is detected using a conventional spiral CT scan.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last1=Donahue|first1=Joseph|last2=Wintermark|first2=Max|date=2015-02-01|title=Perfusion CT and acute stroke imaging: Foundations, applications, and literature review|journal=Journal of Neuroradiology|language=en|volume=42|issue=1|pages=21–29|doi=10.1016/j.neurad.2014.11.003|issn=0150-9861|pmid=25636991}}&lt;/ref&gt; This is better for [[w:stroke|stroke]] diagnosis than other CT types.&lt;ref name=&quot;:1&quot; /&gt;

==== Cardiac Imaging ====
[[File:SADDLE PE.JPG|thumb|CTPA, demonstrating a saddle [[w:pulmonary embolism|embolus]] (dark horizontal line) occluding the [[w:pulmonary artery|pulmonary arteries]] (bright white triangle)|191x191px]]A CT scan of the heart is performed to gain knowledge about cardiac or coronary anatomy.&lt;ref&gt;{{Cite web|url=https://www.nhlbi.nih.gov/health/health-topics/topics/ct|title=Cardiac CT Scan – NHLBI, NIH|website=www.nhlbi.nih.gov|archive-url=https://web.archive.org/web/20171201032800/https://www.nhlbi.nih.gov/health/health-topics/topics/ct|archive-date=2017-12-01|access-date=2017-11-22|url-status=live}}&lt;/ref&gt; Traditionally, cardiac CT scans are used to detect, diagnose, or follow up [[w:coronary artery disease|coronary artery disease]].&lt;ref name=&quot;Wichmann&quot;&gt;{{Cite web|url=https://radiopaedia.org/articles/cardiac-ct-1|title=Cardiac CT {{!}} Radiology Reference Article {{!}} Radiopaedia.org|last=Wichmann|first=Julian L.|website=radiopaedia.org|archive-url=https://web.archive.org/web/20171201040626/https://radiopaedia.org/articles/cardiac-ct-1|archive-date=2017-12-01|access-date=2017-11-22|url-status=dead}}&lt;/ref&gt; More recently CT has played a key role in the fast-evolving field of [[w:Interventional cardiology|transcatheter structural heart interventions]], more specifically in the transcatheter repair and replacement of heart valves.&lt;ref&gt;{{Cite journal|last1=Marwan|first1=Mohamed|last2=Achenbach|first2=Stephan|date=February 2016|title=Role of Cardiac CT Before Transcatheter Aortic Valve Implantation (TAVI)|journal=Current Cardiology Reports|volume=18|issue=2|pages=21|doi=10.1007/s11886-015-0696-3|issn=1534-3170|pmid=26820560|s2cid=41535442}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Moss|first1=Alastair J.|last2=Dweck|first2=Marc R.|last3=Dreisbach|first3=John G.|last4=Williams|first4=Michelle C.|last5=Mak|first5=Sze Mun|last6=Cartlidge|first6=Timothy|last7=Nicol|first7=Edward D.|last8=Morgan-Hughes|first8=Gareth J.|date=2016-11-01|title=Complementary role of cardiac CT in the assessment of aortic valve replacement dysfunction|journal=Open Heart|volume=3|issue=2|pages=e000494|doi=10.1136/openhrt-2016-000494|issn=2053-3624|pmc=5093391|pmid=27843568}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Thériault-Lauzier|first1=Pascal|last2=Spaziano|first2=Marco|last3=Vaquerizo|first3=Beatriz|last4=Buithieu|first4=Jean|last5=Martucci|first5=Giuseppe|last6=Piazza|first6=Nicolo|date=September 2015|title=Computed Tomography for Structural Heart Disease and Interventions|journal=Interventional Cardiology Review|volume=10|issue=3|pages=149–154|doi=10.15420/ICR.2015.10.03.149|issn=1756-1477|pmc=5808729|pmid=29588693}}&lt;/ref&gt; The main forms of cardiac CT scanning are:

[[w:Coronary CT angiography|Coronary CT angiography]] (CCTA): the use of CT to assess the [[w:coronary artery|coronary arteries]] of the [[w:heart|heart]]. The subject receives an [[w:intravenous injection|intravenous injection]] of [[w:radiocontrast|radiocontrast]], and then the heart is scanned using a high-speed CT scanner, allowing radiologists to assess the extent of occlusion in the coronary arteries, usually to diagnose coronary artery disease.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=eR5USB6sRU4C&amp;q=ct+angiography|title=Multidetector-Row CT Angiography|last=Passariello|first=Roberto|date=2006-03-30|publisher=Springer Science &amp; Business Media|isbn=978-3-540-26984-7|language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.radiologyinfo.org/en/info.cfm?pg=angiocoroct|title=Coronary Computed Tomography Angiography (CCTA)|last=Radiology (ACR)|first=Radiological Society of North America (RSNA) and American College of|website=www.radiologyinfo.org|language=en|access-date=2021-03-19}}&lt;/ref&gt;

[[w:Coronary CT calcium scan|Coronary CT calcium scan]]: also used for the assessment of severity of coronary artery disease.&lt;ref&gt;{{Cite journal|last=Greenland|first=Philip|date=2004-01-14|title=Coronary Artery Calcium Score Combined With Framingham Score for Risk Prediction in Asymptomatic Individuals|url=http://jama.jamanetwork.com/article.aspx?doi=10.1001/jama.291.2.210|journal=JAMA|language=en|volume=291|issue=2|pages=210|doi=10.1001/jama.291.2.210|issn=0098-7484}}&lt;/ref&gt; Specifically, it looks for calcium deposits in the coronary arteries that can narrow arteries and increase the risk of a heart attack.&lt;ref&gt;{{Cite journal|last=Gupta|first=Amit|last2=Bera|first2=Kaustav|last3=Kikano|first3=Elias|last4=Pierce|first4=Jonathan D.|last5=Gan|first5=Jonathan|last6=Rajdev|first6=Maharshi|last7=Ciancibello|first7=Leslie M.|last8=Gupta|first8=Aekta|last9=Rajagopalan|first9=Sanjay|date=2022-07|title=Coronary Artery Calcium Scoring: Current Status and Future Directions|url=http://pubs.rsna.org/doi/10.1148/rg.210122|journal=RadioGraphics|language=en|volume=42|issue=4|pages=947–967|doi=10.1148/rg.210122|issn=0271-5333}}&lt;/ref&gt; A typical coronary CT calcium scan is done without the use of radiocontrast, but it can possibly be done from contrast-enhanced images as well.&lt;ref name=&quot;van der BijlJoemai2010&quot;&gt;{{Cite journal|last1=van der Bijl|first1=Noortje|last2=Joemai|first2=Raoul M. S.|last3=Geleijns|first3=Jacob|last4=Bax|first4=Jeroen J.|last5=Schuijf|first5=Joanne D.|last6=de Roos|first6=Albert|last7=Kroft|first7=Lucia J. M.|year=2010|title=Assessment of Agatston Coronary Artery Calcium Score Using Contrast-Enhanced CT Coronary Angiography|journal=American Journal of Roentgenology|volume=195|issue=6|pages=1299–1305|doi=10.2214/AJR.09.3734|issn=0361-803X|pmid=21098187}}&lt;/ref&gt; 

To better visualize the anatomy, post-processing of the images is common.&lt;ref name=&quot;Wichmann&quot; /&gt; Most common are multiplanar reconstructions (MPR) and [[w:volume rendering|volume rendering]]. For more complex anatomies and procedures, such as heart valve interventions, a true [[w:3D reconstruction|3D reconstruction]] or a 3D print is created based on these CT images to gain a deeper understanding.&lt;ref&gt;{{Cite journal|last1=Vukicevic|first1=Marija|last2=Mosadegh|first2=Bobak|last3=Min|first3=James K.|last4=Little|first4=Stephen H.|date=February 2017|title=Cardiac 3D Printing and its Future Directions|journal=JACC: Cardiovascular Imaging|volume=10|issue=2|pages=171–184|doi=10.1016/j.jcmg.2016.12.001|issn=1876-7591|pmc=5664227|pmid=28183437}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Wang|first1=D. D.|last2=Eng|first2=M.|last3=Greenbaum|first3=A.|last4=Myers|first4=E.|last5=Forbes|first5=M.|last6=Pantelic|first6=M.|last7=Song|first7=T.|last8=Nelson|first8=C.|last9=Divine|first9=G.|year=2016|title=Innovative Mitral Valve Treatment with 3D Visualization at Henry Ford|url=http://www.materialise.com/en/blog/innovative-mitral-valve-treatment-3d-visualization-at-henry-ford|journal=JACC: Cardiovascular Imaging|volume=9|issue=11|pages=1349–1352|doi=10.1016/j.jcmg.2016.01.017|pmc=5106323|pmid=27209112|archive-url=https://web.archive.org/web/20171201043336/http://www.materialise.com/en/blog/innovative-mitral-valve-treatment-3d-visualization-at-henry-ford|archive-date=2017-12-01|access-date=2017-11-22|last10=Taylor|first10=A.|last11=Wyman|first11=J.|last12=Guerrero|first12=M.|last13=Lederman|first13=R. J.|last14=Paone|first14=G.|last15=O'Neill|first15=W.|url-status=dead}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Wang|first1=Dee Dee|last2=Eng|first2=Marvin|last3=Greenbaum|first3=Adam|last4=Myers|first4=Eric|last5=Forbes|first5=Michael|last6=Pantelic|first6=Milan|last7=Song|first7=Thomas|last8=Nelson|first8=Christina|last9=Divine|first9=George|date=November 2016|title=Predicting LVOT Obstruction After TMVR|journal=JACC: Cardiovascular Imaging|volume=9|issue=11|pages=1349–1352|doi=10.1016/j.jcmg.2016.01.017|issn=1876-7591|pmc=5106323|pmid=27209112}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Jacobs|first1=Stephan|last2=Grunert|first2=Ronny|last3=Mohr|first3=Friedrich W.|last4=Falk|first4=Volkmar|date=February 2008|title=3D-Imaging of cardiac structures using 3D heart models for planning in heart surgery: a preliminary study|journal=Interactive Cardiovascular and Thoracic Surgery|volume=7|issue=1|pages=6–9|doi=10.1510/icvts.2007.156588|issn=1569-9285|pmid=17925319|doi-access=free}}&lt;/ref&gt;

=== Interventional ===
CT-guided interventional procedures involve minimally invasive techniques guided by computed tomography imaging. These procedures utilize detailed cross-sectional images generated by CT scans to precisely guide various interventions. Common interventions performed under CT-guidance include biopsies for diagnostic purposes, drainage of fluid-filled areas, radiofrequency ablation to destroy tumors, and procedures like vertebroplasty or kyphoplasty for stabilizing fractured vertebrae.&lt;ref&gt;{{Cite journal|last=Tsalafoutas|first=Ioannis A.|last2=Tsapaki|first2=Virginia|last3=Triantopoulou|first3=Charicleia|last4=Gorantonaki|first4=Akrivi|last5=Papailiou|first5=John|date=2007-06|title=CT-Guided Interventional Procedures without CT Fluoroscopy Assistance: Patient Effective Dose and Absorbed Dose Considerations|url=https://www.ajronline.org/doi/10.2214/AJR.06.0705|journal=American Journal of Roentgenology|language=en|volume=188|issue=6|pages=1479–1484|doi=10.2214/AJR.06.0705|issn=0361-803X}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Seong|first=Ju-Yong|last2=Kim|first2=Jin-Sung|last3=Jung|first3=Byungjoo|last4=Lee|first4=Sang-Ho|last5=Kang|first5=Ho Yeong|date=2009|title=CT-Guided Percutaneous Vertebroplasty in the Treatment of an Upper Thoracic Compression Fracture|url=https://www.kjronline.org/DOIx.php?id=10.3348/kjr.2009.10.2.185|journal=Korean Journal of Radiology|volume=10|issue=2|pages=185|doi=10.3348/kjr.2009.10.2.185|issn=1229-6929|pmc=PMC2651434|pmid=19270865}}&lt;/ref&gt; The real-time imaging provided by CT ensures accuracy in needle or catheter placement during these procedures.&lt;ref&gt;{{Cite journal|last=Lucey|first=Brian C.|last2=Varghese|first2=Jose C.|last3=Hochberg|first3=Aaron|last4=Blake|first4=Michael A.|last5=Soto|first5=Jorge A.|date=2007-05|title=CT-Guided Intervention with Low Radiation Dose: Feasibility and Experience|url=https://www.ajronline.org/doi/10.2214/AJR.06.0378|journal=American Journal of Roentgenology|language=en|volume=188|issue=5|pages=1187–1194|doi=10.2214/AJR.06.0378|issn=0361-803X}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Kliger|first=Chad|last2=Jelnin|first2=Vladimir|last3=Sharma|first3=Sonnit|last4=Panagopoulos|first4=Georgia|last5=Einhorn|first5=Bryce N.|last6=Kumar|first6=Robert|last7=Cuesta|first7=Francisco|last8=Maranan|first8=Leandro|last9=Kronzon|first9=Itzhak|date=2014-02|title=CT Angiography–Fluoroscopy Fusion Imaging for Percutaneous Transapical Access|url=https://linkinghub.elsevier.com/retrieve/pii/S1936878X1300822X|journal=JACC: Cardiovascular Imaging|language=en|volume=7|issue=2|pages=169–177|doi=10.1016/j.jcmg.2013.10.009}}&lt;/ref&gt; CT fluoroscopy proves to be a valuable clinical instrument, enhancing the efficiency of percutaneous abdominal and pelvic interventional procedures.&lt;ref&gt;{{Cite journal|last=Daly|first=Barry|last2=Templeton|first2=Philip A.|date=1999-05|title=Real-time CT Fluoroscopy: Evolution of an Interventional Tool|url=http://pubs.rsna.org/doi/10.1148/radiology.211.2.r99ma51309|journal=Radiology|language=en|volume=211|issue=2|pages=309–315|doi=10.1148/radiology.211.2.r99ma51309|issn=0033-8419}}&lt;/ref&gt; 

CT-guided biopsies encompass guided imaging techniques, employing a CT scanner to direct the needle insertion during the procedure. These guided procedures can be either diagnostic or therapeutic.&lt;ref&gt;{{Cite book|title=Percutaneous image-guided biopsy|last=Ahrar|first=Kamran|last2=Gupta|first2=Sanjay|date=2014|publisher=Springer|isbn=978-1-4614-8217-8|location=New York}} Page no. 19&lt;/ref&gt;&lt;ref&gt;{{Cite book|title=CT- and MR-guided interventions in radiology|last=Mahnken|first=Andreas H.|last2=Wilhelm|first2=Kai|last3=Ricke|first3=Jens|date=2013|publisher=Springer|isbn=978-3-642-33581-5|edition=2nd ed|location=Berlin New York}}&lt;/ref&gt; CT-guided biopsies are widely utilized for diagnosing hepatic, renal, pulmonary, bone, pancreatic, adrenal, lymphatic, and brain lesions.&lt;ref&gt;{{Cite journal|last=Hyun|first=Kwon H.A.|last2=Sachs|first2=Peter B.|last3=Haaga|first3=John R.|last4=Abdul-Karim|first4=Fadi|date=1991-04|title=CT-guided liver biopsy: An update|url=https://doi.org/10.1016/0899-7071(91)90155-O|journal=Clinical Imaging|volume=15|issue=2|pages=99–104|doi=10.1016/0899-7071(91)90155-o|issn=0899-7071}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Uppot|first=Raul N.|last2=Harisinghani|first2=Mukesh G.|last3=Gervais|first3=Debra A.|date=2010-06|title=Imaging-Guided Percutaneous Renal Biopsy: Rationale and Approach|url=https://www.ajronline.org/doi/10.2214/AJR.10.4427|journal=American Journal of Roentgenology|language=en|volume=194|issue=6|pages=1443–1449|doi=10.2214/AJR.10.4427|issn=0361-803X}}&lt;/ref&gt;

CT-guided nephrostomies demonstrate feasibility and effectiveness, particularly in instances of iatrogenic ureteral injury.&lt;ref&gt;{{Cite journal|last=Jiao|first=Dechao|last2=Li|first2=Zongming|last3=Li|first3=Zhiguo|last4=Shui|first4=Shaofeng|last5=Han|first5=Xin-wei|date=2017-08-01|title=Flat detector cone beam CT-guided nephrostomy using virtual navigation in patients with iatrogenic ureteral injury|url=https://doi.org/10.1007/s11547-017-0751-9|journal=La radiologia medica|language=en|volume=122|issue=8|pages=557–563|doi=10.1007/s11547-017-0751-9|issn=1826-6983}}&lt;/ref&gt; This technique enables precise needle placement and the detection of subtle density variations within tissues&lt;ref&gt;{{Cite journal|last=Haaga|last2=Zelch|first2=Mg|last3=Alfidi|first3=Rj|last4=Stewart|first4=Bh|last5=Daugherty|first5=Jd|date=1977-04-01|title=CT-guided antegrade pyelography and percutaneous nephrostomy|url=https://www.ajronline.org/doi/10.2214/ajr.128.4.621|journal=American Journal of Roentgenology|language=en|volume=128|issue=4|pages=621–624|doi=10.2214/ajr.128.4.621|issn=0361-803X}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Smith|first=Paul Edmund|last2=Luong|first2=Ian Thuan Hien|last3=van der Vliet|first3=Andrew Hans|date=2018-08|title=CT ‐guided nephrostomy: Re‐inventing the wheel for the occasional interventionalist|url=https://onlinelibrary.wiley.com/doi/10.1111/1754-9485.12720|journal=Journal of Medical Imaging and Radiation Oncology|language=en|volume=62|issue=4|pages=520–524|doi=10.1111/1754-9485.12720|issn=1754-9477}}&lt;/ref&gt; CT-guided percutaneous nephrostomy has proven to be efficient and safe, associated with low complication rates&lt;ref&gt;{{Cite journal|last=Egilmez|first=H.|last2=Oztoprak|first2=I.|last3=Atalar|first3=M.|last4=Cetin|first4=A.|last5=Gumus|first5=C.|last6=Gultekin|first6=Y.|last7=Bulut|first7=S.|last8=Arslan|first8=M.|last9=Solak|first9=O.|date=2007-09|title=The place of computed tomography as a guidance modality in percutaneous nephrostomy: analysis of a 10-year single-center experience|url=http://journals.sagepub.com/doi/10.1080/02841850701416528|journal=Acta Radiologica|language=en|volume=48|issue=7|pages=806–813|doi=10.1080/02841850701416528|issn=0284-1851}}&lt;/ref&gt;.

CT-guided tumor ablation involves using CT imaging for precise guidance during minimally invasive procedures to treat tumors. Techniques such as radiofrequency ablation  and microwave ablation utilize heat to destroy cancerous tissues.&lt;ref&gt;{{Cite journal|last=Engstrand|first=Jennie|last2=Toporek|first2=Grzegorz|last3=Harbut|first3=Piotr|last4=Jonas|first4=Eduard|last5=Nilsson|first5=Henrik|last6=Freedman|first6=Jacob|date=2017-01|title=Stereotactic CT-Guided Percutaneous Microwave Ablation of Liver Tumors With the Use of High-Frequency Jet Ventilation: An Accuracy and Procedural Safety Study|url=https://www.ajronline.org/doi/10.2214/AJR.15.15803|journal=American Journal of Roentgenology|language=en|volume=208|issue=1|pages=193–200|doi=10.2214/AJR.15.15803|issn=0361-803X}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Kulkarni|first=Suyash S|last2=Shetty|first2=Nitin S|last3=Polnaya|first3=Ashwin M|last4=Janu|first4=Amit|last5=Kumar|first5=Suresh|last6=Puri|first6=Ajay|last7=Gulia|first7=Ashish|last8=Rangarajan|first8=Venkatesh|date=2017-07|title=CT-guided radiofrequency ablation in osteoid osteoma: Result from a tertiary cancer centre in India|url=http://www.thieme-connect.de/DOI/DOI?10.4103/ijri.IJRI_30_17|journal=Indian Journal of Radiology and Imaging|language=en|volume=27|issue=03|pages=318–323|doi=10.4103/ijri.IJRI_30_17|issn=0971-3026}}&lt;/ref&gt; The visualization capabilities of CT enables to accurately target and ablate tumors, offering a less invasive alternative for patients with conditions such as liver tumors.&lt;ref&gt;{{Cite journal|last=Raissi|first=Driss|last2=Sanampudi|first2=Sreeja|last3=Yu|first3=Qian|last4=Winkler|first4=Michael|date=2022-01-20|title=CT-guided microwave ablation of hepatic malignancies via transpulmonary approach without ancillary techniques|url=https://clinicalimagingscience.org/ct-guided-microwave-ablation-of-hepatic-malignancies-via-transpulmonary-approach-without-ancillary-techniques/|journal=Journal of Clinical Imaging Science|language=en|volume=12|pages=2|doi=10.25259/JCIS_152_2021|issn=2156-5597|pmc=PMC8813600|pmid=35127245}}&lt;/ref&gt;

=== Vascular Imaging===
[[w:Computed tomography angiography|Computed tomography angiography]] (CTA) is a type of [[w:contrast CT|contrast CT]] to visualize the [[w:arteries|arteries]] and [[w:vein|vein]]s throughout the body.&lt;ref&gt;{{Citation |last1=McDermott |first1=M. |title=Chapter 10 – Critical care in acute ischemic stroke |date=2017-01-01 |journal=Handbook of Clinical Neurology |volume=140 |pages=153–176 |editor-last=Wijdicks |editor-first=Eelco F. M. |series=Critical Care Neurology Part I |publisher=Elsevier |language=en |doi=10.1016/b978-0-444-63600-3.00010-6 |pmid=28187798 |last2=Jacobs |first2=T. |last3=Morgenstern |first3=L. |editor2-last=Kramer |editor2-first=Andreas H.}}&lt;/ref&gt;  This ranges from arteries serving the [[w:brain|brain]] to those bringing blood to the [[w:lung|lung]]s, [[w:kidney|kidney]]s, [[w:arm|arm]]s and [[w:leg|leg]]s. An example of this type of exam is [[w:CT pulmonary angiogram|CT pulmonary angiogram]] (CTPA) used to diagnose [[w:pulmonary embolism|pulmonary embolism]] (PE). It employs computed tomography and an [[w:iodinated contrast|iodine-based contrast agent]] to obtain an image of the [[w:pulmonary artery|pulmonary arteries]].&lt;ref&gt;{{Cite journal |last1=Zeman |first1=R K |last2=Silverman |first2=P M |last3=Vieco |first3=P T |last4=Costello |first4=P |date=1995-11-01 |title=CT angiography. |journal=American Journal of Roentgenology |volume=165 |issue=5 |pages=1079–1088 |doi=10.2214/ajr.165.5.7572481 |issn=0361-803X |pmid=7572481 |doi-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last1=Ramalho |first1=Joana |url=https://books.google.com/books?id=FKdMAgAAQBAJ&amp;q=cta+is+an+imaging |title=Vascular Imaging of the Central Nervous System: Physical Principles, Clinical Applications, and Emerging Techniques |last2=Castillo |first2=Mauricio |date=2014-03-31 |publisher=John Wiley &amp; Sons |isbn=978-1-118-18875-0 |page=69 |language=en}}&lt;/ref&gt;

== Other uses ==
Industrial CT scanning (industrial computed tomography) is a process which utilizes X-ray equipment to produce 3D representations of components both externally and internally. Industrial CT scanning has been utilized in many areas of industry for internal inspection of components. Some of the key uses for CT scanning have been flaw detection, failure analysis, metrology, assembly analysis, image-based finite element methods&lt;ref&gt;{{Cite journal|last1=Evans|first1=Ll. M.|last2=Margetts|first2=L.|last3=Casalegno|first3=V.|last4=Lever|first4=L. M.|last5=Bushell|first5=J.|last6=Lowe|first6=T.|last7=Wallwork|first7=A.|last8=Young|first8=P.|last9=Lindemann|first9=A.|date=2015-05-28|title=Transient thermal finite element analysis of CFC–Cu ITER monoblock using X-ray tomography data|url=https://www.researchgate.net/publication/277338941|journal=Fusion Engineering and Design|volume=100|pages=100–111|doi=10.1016/j.fusengdes.2015.04.048|archive-url=https://web.archive.org/web/20151016091649/http://www.researchgate.net/publication/277338941_Transient_thermal_finite_element_analysis_of_CFCCu_ITER_monoblock_using_X-ray_tomography_data|archive-date=2015-10-16|doi-access=free|url-status=live}}&lt;/ref&gt; and reverse engineering applications. CT scanning is also employed in the imaging and conservation of museum artifacts.&lt;ref&gt;{{Cite journal|last=Payne, Emma Marie|year=2012|title=Imaging Techniques in Conservation|url=http://discovery.ucl.ac.uk/1443164/1/56-566-2-PB.pdf|journal=Journal of Conservation and Museum Studies|volume=10|issue=2|pages=17–29|doi=10.5334/jcms.1021201|doi-access=free}}&lt;/ref&gt;

CT scanning has also found an application in transport security (predominantly airport security) where it is currently used in a materials analysis context for explosives detection CTX (explosive-detection device)&lt;ref&gt;{{Cite book|title=Anomaly Detection and Imaging with X-Rays (ADIX) III|last1=P. Babaheidarian|last2=D. Castanon|date=2018|isbn=978-1-5106-1775-9|editor-last1=Greenberg|editor-first1=Joel A.|pages=12|chapter=Joint reconstruction and material classification in spectral CT|doi=10.1117/12.2309663|editor-last2=Gehm|editor-first2=Michael E.|editor-last3=Neifeld|editor-first3=Mark A.|editor-last4=Ashok|editor-first4=Amit|s2cid=65469251}}&lt;/ref&gt;&lt;ref name=&quot;jin12securityct&quot;&gt;{{Cite book|title=Second International Conference on Image Formation in X-Ray Computed Tomography|last1=P. Jin|last2=E. Haneda|last3=K. D. Sauer|last4=C. A. Bouman|date=June 2012|chapter=A model-based 3D multi-slice helical CT reconstruction algorithm for transportation security application|access-date=2015-04-05|chapter-url=https://engineering.purdue.edu/~bouman/publications/orig-pdf/CT-2012a.pdf|archive-url=https://web.archive.org/web/20150411000659/https://engineering.purdue.edu/~bouman/publications/orig-pdf/CT-2012a.pdf|archive-date=2015-04-11|url-status=dead}}&lt;/ref&gt;&lt;ref name=&quot;jin12securityctprior&quot;&gt;{{Cite book|title=Signals, Systems and Computers (ASILOMAR), 2012 Conference Record of the Forty Sixth Asilomar Conference on|last1=P. Jin|last2=E. Haneda|last3=C. A. Bouman|date=November 2012|publisher=IEEE|pages=613–636|chapter=Implicit Gibbs prior models for tomographic reconstruction|access-date=2015-04-05|chapter-url=https://engineering.purdue.edu/~bouman/publications/pdf/Asilomar-2012-Pengchong.pdf|archive-url=https://web.archive.org/web/20150411025559/https://engineering.purdue.edu/~bouman/publications/pdf/Asilomar-2012-Pengchong.pdf|archive-date=2015-04-11|url-status=dead}}&lt;/ref&gt;&lt;ref name=&quot;kisner13securityct&quot;&gt;{{Cite book|title=Security Technology (ICCST), 2013 47th International Carnahan Conference on|last1=S. J. Kisner|last2=P. Jin|last3=C. A. Bouman|last4=K. D. Sauer|last5=W. Garms|last6=T. Gable|last7=S. Oh|last8=M. Merzbacher|last9=S. Skatter|date=October 2013|publisher=IEEE|chapter=Innovative data weighting for iterative reconstruction in a helical CT security baggage scanner|access-date=2015-04-05|chapter-url=https://engineering.purdue.edu/~bouman/publications/pdf/iccst2013.pdf|archive-url=https://web.archive.org/web/20150410234541/https://engineering.purdue.edu/~bouman/publications/pdf/iccst2013.pdf|archive-date=2015-04-10|url-status=dead}}&lt;/ref&gt; and is also under consideration for automated baggage/parcel security scanning using computer vision based object recognition algorithms that target the detection of specific threat items based on 3D appearance (e.g. guns, knives, liquid containers).&lt;ref name=&quot;megherbi10baggage&quot;&gt;{{Cite journal|last=Megherbi|first=Najla|last2=Flitton|first2=Greg T.|last3=Breckon|first3=Toby P.|date=2010-09|title=A classifier based approach for the detection of potential threats in CT based Baggage Screening|url=https://ieeexplore.ieee.org/document/5653676/|journal=2010 IEEE International Conference on Image Processing|pages=1833–1836|doi=10.1109/ICIP.2010.5653676}}&lt;/ref&gt;&lt;ref name=&quot;megherbi12baggage&quot;&gt;{{Cite journal|last=Megherbi|first=Najla|last2=Han|first2=Jiwan|last3=Breckon|first3=Toby P.|last4=Flitton|first4=Greg T.|date=2012-09|title=A comparison of classification approaches for threat detection in CT based baggage screening|url=https://ieeexplore.ieee.org/document/6467558/|journal=2012 19th IEEE International Conference on Image Processing|pages=3109–3112|doi=10.1109/ICIP.2012.6467558}}&lt;/ref&gt;&lt;ref name=&quot;flitton13interestpoint&quot;&gt;{{Cite journal|last=Flitton|first=Greg|last2=Breckon|first2=Toby P.|last3=Megherbi|first3=Najla|date=2013-09|title=A comparison of 3D interest point descriptors with application to airport baggage object detection in complex CT imagery|url=https://doi.org/10.1016/j.patcog.2013.02.008|journal=Pattern Recognition|volume=46|issue=9|pages=2420–2436|doi=10.1016/j.patcog.2013.02.008|issn=0031-3203}}&lt;/ref&gt; 

X-ray CT is used in geological studies to quickly reveal materials inside a drill core.&lt;ref&gt;{{Cite web|url=http://www.jamstec.go.jp/chikyu/e/about/laboratory.html|title=Laboratory {{!}} About Chikyu {{!}} The Deep-sea Scientific Drilling Vessel CHIKYU|website=www.jamstec.go.jp|access-date=2019-10-24}}&lt;/ref&gt; Dense minerals such as pyrite and barite appear brighter and less dense components such as clay appear dull in CT images.&lt;ref&gt;{{Cite journal|last1=Tonai|first1=Satoshi|last2=Kubo|first2=Yusuke|last3=Tsang|first3=Man-Yin|last4=Bowden|first4=Stephen|last5=Ide|first5=Kotaro|last6=Hirose|first6=Takehiro|last7=Kamiya|first7=Nana|last8=Yamamoto|first8=Yuzuru|last9=Yang|first9=Kiho|date=2019|title=A New Method for Quality Control of Geological Cores by X-Ray Computed Tomography: Application in IODP Expedition 370|journal=Frontiers in Earth Science|language=English|volume=7|doi=10.3389/feart.2019.00117|issn=2296-6463|doi-access=free|last10=Yamada|first10=Yasuhiro|last11=Morono|first11=Yuki|s2cid=171394807}}&lt;/ref&gt;

X-ray CT and micro-CT can also be used for the conservation and preservation of objects of cultural heritage. For many fragile objects, direct research and observation can be damaging and can degrade the object over time. Using CT scans, conservators and researchers are able to determine the material composition of the objects they are exploring, such as the position of ink along the layers of a scroll, without any additional harm. After scanning these objects, computational methods can be employed to examine the insides of these objects.&lt;ref&gt;{{Cite journal|last1=Seales|first1=W. B.|last2=Parker|first2=C. S.|last3=Segal|first3=M.|last4=Tov|first4=E.|last5=Shor|first5=P.|last6=Porath|first6=Y.|year=2016|title=From damage to discovery via virtual unwrapping: Reading the scroll from En-Gedi|journal=Science Advances|volume=2|issue=9|pages=e1601247|bibcode=2016SciA....2E1247S|doi=10.1126/sciadv.1601247|issn=2375-2548|pmc=5031465|pmid=27679821}}&lt;/ref&gt; Micro-CT has also proved useful for analyzing more recent artifacts such as still-sealed historic correspondence that employed the technique of letterlocking (complex folding and cuts) that provided a &quot;tamper-evident locking mechanism&quot;.&lt;ref&gt;{{Cite web|url=https://www.wsj.com/articles/a-letter-sealed-for-centuries-has-been-readwithout-even-opening-it-11614679203|title=A Letter Sealed for Centuries Has Been Read—Without Even Opening It|last=Castellanos|first=Sara|date=2 March 2021|website=The Wall Street Journal|access-date=2 March 2021}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Dambrogio|first1=Jana|last2=Ghassaei|first2=Amanda|last3=Staraza Smith|first3=Daniel|last4=Jackson|first4=Holly|last5=Demaine|first5=Martin L.|date=2 March 2021|title=Unlocking history through automated virtual unfolding of sealed documents imaged by X-ray microtomography|journal=Nature Communications|volume=12|issue=1|page=1184|bibcode=2021NatCo..12.1184D|doi=10.1038/s41467-021-21326-w|pmc=7925573|pmid=33654094}}&lt;/ref&gt;

== Procedure ==
Before starting the procedure, the patient preparation is necessary to ensure optimal scan quality and safety. This preparation includes a thorough examination of the patient's medical history to identify any potential contraindications. The patient is briefed about the procedure, and informed written consent is obtained from the patient on family member.[[File:CT ScoutView.jpg|thumb|206x206px|Topogram ]]The specific preparation measures vary depending on the type of scan and the targeted organ. For abdominal or pelvic CT scans, fasting is essential to minimize interference from bowel gas and enhance the visualization of organs. Pre-scan instructions are also influenced by the use of contrast material, with some patients advised to refrain from certain medications, especially those affecting kidney function.

Patients undergoing CT scans may experience anxiety, either due to the unfamiliar environment or, in some cases, claustrophobia.&lt;ref&gt;{{Cite journal|last=Heyer|first=Christoph M.|last2=Thüring|first2=Johannes|last3=Lemburg|first3=Stefan P.|last4=Kreddig|first4=Nina|last5=Hasenbring|first5=Monika|last6=Dohna|first6=Martha|last7=Nicolas|first7=Volkmar|date=2015-01|title=Anxiety of Patients Undergoing CT Imaging—An Underestimated Problem?|url=https://linkinghub.elsevier.com/retrieve/pii/S1076633214002980|journal=Academic Radiology|language=en|volume=22|issue=1|pages=105–112|doi=10.1016/j.acra.2014.07.014}}&lt;/ref&gt; Consequently, maintaining stillness during the examination can be challenging for them. In such cases, commonly in children, a sedative can be employed to alleviate the patient's anxiety and ensure a smoother scanning process.&lt;ref&gt;{{Cite journal|last=Keeter|first=S|last2=Benator|first2=R M|last3=Weinberg|first3=S M|last4=Hartenberg|first4=M A|date=1990-06|title=Sedation in pediatric CT: national survey of current practice.|url=http://pubs.rsna.org/doi/10.1148/radiology.175.3.2343126|journal=Radiology|language=en|volume=175|issue=3|pages=745–752|doi=10.1148/radiology.175.3.2343126|issn=0033-8419}}&lt;/ref&gt;

Before the actual scan, a topogram, also known as a scout image or localizer image, is taken which is a low-dose, low-resolution radiographic image. This initial image helps to plan the coverage and orientation of the subsequent CT scan. The topogram provides a preliminary overview of the area to be imaged, allowing technologist to plan the scan.&lt;ref&gt;{{Cite journal|last=Li|first=Baojun|last2=Behrman|first2=Richard H.|last3=Norbash|first3=Alexander M.|date=2012-06|title=Comparison of topogram‐based body size indices for CT dose consideration and scan protocol optimization|url=https://aapm.onlinelibrary.wiley.com/doi/10.1118/1.4718569|journal=Medical Physics|language=en|volume=39|issue=6Part1|pages=3456–3465|doi=10.1118/1.4718569|issn=0094-2405}}&lt;/ref&gt; Since, Topograms have a larger field of view than main scan, they can also play a role in revealing significant findings outside the scan field of view.&lt;ref&gt;{{Cite journal|last=Lee|first=Matthew H.|last2=Lubner|first2=Meghan G.|last3=Mellnick|first3=Vincent M.|last4=Menias|first4=Christine O.|last5=Bhalla|first5=Sanjeev|last6=Pickhardt|first6=Perry J.|date=2021-10|title=The CT scout view: complementary value added to abdominal CT interpretation|url=https://link.springer.com/10.1007/s00261-021-03135-3|journal=Abdominal Radiology|language=en|volume=46|issue=10|pages=5021–5036|doi=10.1007/s00261-021-03135-3|issn=2366-004X}}&lt;/ref&gt;

=== Non-contrast CT ===
[[File:CT ABDOMEN.jpg|thumb|194x194px|Non Contrast CT Abdomen]]
CT procedure in which contrast media is not used is often called as Non-Contrast CT (NCCT) or plain CT. This procedure is employed when there is already a sufficient contrast distinction in the target tissues, rendering the resulting image diagnostically significant. The process involves acquiring a topogram, followed by scanning the region of interest and reconstructing the data, marking the conclusion of the procedure. The non-contrast CT scans are rapid, less hazardous, and cost-effective procedures. Non-contrast CT head scans play a pivotal role in the identification of various conditions, encompassing traumatic hemorrhages, subdural hematomas, cerebral edema, fractures, and in detecting  foreign bodies, such as tempered glass, which might be overlooked&lt;ref&gt;{{Cite journal|last=Faheem|first=Mohd|last2=Kumar|first2=Raj|last3=Jaiswal|first3=Manish|last4=Ansari|first4=Mohammad Ahmed|last5=Saba|first5=Noor us|date=2019-12|title=“Mosaic Pattern” Foreign Bodies in Computed Tomography of the Head: A Specific Sign to Detect Tempered Glass-Related Head Injury|url=http://www.thieme-connect.de/DOI/DOI?10.1055/s-0039-1700302|journal=Indian Journal of Neurosurgery|language=en|volume=08|issue=03|pages=193–195|doi=10.1055/s-0039-1700302|issn=2277-954X}}&lt;/ref&gt;.   

=== Contrast CT ===
[[File:Normal contrast enhanced abdominal CT.jpg|thumb|Normal contrast enhanced abdominal CT.]]
Contrast CT or CECT procedures invole the use of a contrast medium for better visualization. Contrast media, also known as contrast agents, are substances used in imaging to improve the visibility of internal structures or fluids during diagnostic procedures. These agents enhance the differentiation between various tissues, and between normal and abnormal tissues allowing for clearer and more detailed imaging.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=xb-xLHTqOi0C&amp;q=contrast+in+ct|title=Fundamentals of Body CT|last1=Webb|first1=Wayne Richard|last2=Brant|first2=William E.|last3=Major|first3=Nancy M.|date=2006-01-01|publisher=Elsevier Health Sciences|isbn=978-1-4160-0030-3|page=168|language=en}}&lt;/ref&gt; Contrast agents employed in CT imaging also know as Radio contrasts are generally categorized into Positive, Negative, and Neutral contrast agents. Positive contrast agents increase the x-ray attenuation and negative contrast agents reduce x-ray attenuation. Neutral contrast media, on the other hand, do not alter attenuation but are employed to enhance distention.&lt;ref&gt;{{Cite journal|last=Callahan|first=Michael J.|last2=Talmadge|first2=Jennifer M.|last3=MacDougall|first3=Robert|last4=Buonomo|first4=Carlo|last5=Taylor|first5=George A.|date=2016-05|title=The Use of Enteric Contrast Media for Diagnostic CT, MRI, and Ultrasound in Infants and Children: A Practical Approach|url=https://www.ajronline.org/doi/10.2214/AJR.15.15437|journal=American Journal of Roentgenology|language=en|volume=206|issue=5|pages=973–979|doi=10.2214/AJR.15.15437|issn=0361-803X}}&lt;/ref&gt;    [[File:CT Contrast classification.png|thumb|CT Contrast classification|231x231px|left]]
Positive contrast agents can be categorized into Iodinated, Oily, or Barium sulfate contrast agents based on their composition. The most prevalent among them are Iodinated contrast agents, which are based on Iodine. These are further classified into Ionic contrast media and non-ionic contrast media.&lt;ref&gt;{{Cite journal|last=Baerlocher|first=M. O.|last2=Asch|first2=M.|last3=Myers|first3=A.|date=2010-04-20|title=The use of contrast media|url=http://www.cmaj.ca/cgi/doi/10.1503/cmaj.090118|journal=Canadian Medical Association Journal|language=en|volume=182|issue=7|pages=697–697|doi=10.1503/cmaj.090118|issn=0820-3946|pmc=PMC2855918|pmid=20231343}}&lt;/ref&gt; The ionic iodinated contrast is further divided into Ionic monomers and Ionic dimers, &amp; the  non ionic iodinated contrast media is divided into non-ionic monomers and non-ionic dimers. Ionic monomers constitute a type of high-osmolar contrast media with an Iodine-to-particle ratio of 3:2, while Ionic dimers and Non-ionic monomers share an Iodine-to-particle ratio of 3:1. Non-ionic dimers, on the other hand, have a ratio of 6:1. The decrease in osmolarity is associated with the increase in the viscosity.&lt;ref&gt;{{Cite journal|last=Bottinor|first=Wendy|last2=Polkampally|first2=Pritam|last3=Jovin|first3=Ion|date=2013-08-16|title=Adverse Reactions to Iodinated Contrast Media|url=http://www.thieme-connect.de/DOI/DOI?10.1055/s-0033-1348885|journal=International Journal of Angiology|language=en|volume=22|issue=03|pages=149–154|doi=10.1055/s-0033-1348885|issn=1061-1711}}&lt;/ref&gt;  

Barium based contrast are used in the imaging of gastrointestinal tract. These contrast agents help outline the contours of the GI organs, such as the stomach and intestines, making them more visible on CT scans. Barium provides a high degree of contrast due to high x ray attenuation. It can also be used for double contrast studies eg in case of barium enema. It is water insoluble and is not absorbed by the gut.        

Mainly Intravenous iodinated contrast media are used as positive contrast agents is used in CT, with some procedures Oral contrast and Negative contrast media can also be used. The use of contrast media mandates a thorough review of the patient's medical history and allergies, &amp;  assessment of renal function. Explicit written consent is imperative before administering the contrast material, delivered through an intravenous line in the arm or hand at a controlled rate via hand injection or a pressure injector. Upon injecting, the scan is initiated with precise timing. This temporal coordination is of paramount importance for observing distinct levels of enhancement throughout the scanning process.    

Contrast phases refer to distinct stages in the enhancement of blood vessels or tissue after the administration of a intravenous contrast agent during a CT procedure. During a contrast-enhanced CT scan, various contrast phases delineate the dynamic enhancement of blood vessels. The early arterial phase manifests 15-20 seconds after injection of contrast. Following this, the late arterial phase transpires 30-40 seconds post-injection. Subsequently, the portal venous phase unfolds 70-90 seconds after injection. The nephrogenic phase starts at 100-120 seconds post-injection. The excretory phase also called as washout phase occurs at 5-10 minutes after injection. 

==== Contrast injection techniques ====
Test bolus is a small, preliminary injection of contrast agent given to a patient before the actual CT scan. The purpose of the test bolus is to determine the optimal timing for the contrast-enhanced scan and in the mean time also assess the integrity of venous access before administering the complete bolus of contrast medium.&lt;ref&gt;{{Cite journal|last=Bae|first=Kyongtae T.|date=2003-06|title=Peak Contrast Enhancement in CT and MR Angiography: When Does It Occur and Why? Pharmacokinetic Study in a Porcine Model|url=http://pubs.rsna.org/doi/10.1148/radiol.2273020102|journal=Radiology|language=en|volume=227|issue=3|pages=809–816|doi=10.1148/radiol.2273020102|issn=0033-8419}}&lt;/ref&gt;

Bolus tracking is a technique employed in CECT to monitor the concentration of contrast material within a designated region of interest, typically a blood vessel. A region of interest (ROI) is typically positioned just before the target organ, The scanning process starts when the contrast concentration attains a predefined HU level, ensuring optimal filling of blood vessels with contrast. This approach aids in acquiring images at the point of peak enhancements.&lt;ref&gt;{{Cite journal|last=Nebelung|first=Heiner|last2=Brauer|first2=Thomas|last3=Seppelt|first3=Danilo|last4=Hoffmann|first4=Ralf-Thorsten|last5=Platzek|first5=Ivan|date=2021-02|title=Coronary computed tomography angiography (CCTA): effect of bolus-tracking ROI positioning on image quality|url=https://link.springer.com/10.1007/s00330-020-07131-x|journal=European Radiology|language=en|volume=31|issue=2|pages=1110–1118|doi=10.1007/s00330-020-07131-x|issn=0938-7994|pmc=PMC7813743|pmid=32809163}}&lt;/ref&gt;

== Mechanism ==
[[File:ct-internals.jpg|thumb|right|CT scanner with cover removed to show internal components. Legend:
&lt;br /&gt;T: X-ray tube
&lt;br /&gt;D: X-ray detectors
&lt;br /&gt;X: X-ray beam
&lt;br /&gt;R: Gantry rotation]]
[[File:Sinogram and sample image of computed tomography of the jaw.jpg|thumb|Left image is a ''sinogram'' which is a graphic representation of the raw data obtained from a CT scan. At right is an image sample derived from the raw data.&lt;ref&gt;{{Cite journal|last1=Jun|first1=Kyungtaek|last2=Yoon|first2=Seokhwan|year=2017|title=Alignment Solution for CT Image Reconstruction using Fixed Point and Virtual Rotation Axis|journal=Scientific Reports|volume=7|pages=41218|arxiv=1605.04833|bibcode=2017NatSR...741218J|doi=10.1038/srep41218|issn=2045-2322|pmc=5264594|pmid=28120881}}&lt;/ref&gt;]]
Computed tomography scanner operates by using an X-ray tube that generates X-rays and rotates around the patient; [[w:X-ray detector|X-ray detector]]s are positioned on the opposite side of the the X-ray source.&lt;ref&gt;{{Cite web|url=https://www.nibib.nih.gov/science-education/science-topics/computed-tomography-ct|title=Computed Tomography (CT)|website=www.nibib.nih.gov|access-date=2021-03-20}}&lt;/ref&gt; As the X-rays pass through the patient, they are attenuated by various tissues according to their density.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=nPisjRy4LNAC&amp;pg=PA3|title=Radiation Exposure and Image Quality in X-Ray Diagnostic Radiology: Physical Principles and Clinical Applications|last1=Aichinger|first1=Horst|last2=Dierker|first2=Joachim|last3=Joite-Barfuß|first3=Sigrid|last4=Säbel|first4=Manfred|date=2011-10-25|publisher=Springer Science &amp; Business Media|isbn=978-3-642-11241-6|pages=5|language=en}}&lt;/ref&gt; Tissues with higher density attenuate more x-ray photons while tissues with low density attenuate less, this attenuation data is acquired by the detectors around the patient.  A visual representation of the raw data obtained is called a sinogram, yet it is not sufficient for interpretation.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=ylcfAQAAMAAJ&amp;q=A+set+of+many+such+projections+under+different+angles+organized+in+2D+is+called+sinogram|title=Statistical Image Reconstruction Algorithms Using Paraboloidal Surrogates for PET Transmission Scans|last=Erdoğan|first=Hakan|date=1999|publisher=University of Michigan|isbn=978-0-599-63374-2|language=en}}&lt;/ref&gt; The term [[wikt:sinogram|sinogram]] was introduced by Paul Edholm and Bertil Jacobson in 1975.&lt;ref&gt;{{Cite journal|last1=Edholm|first1=Paul|last2=Gabor|first2=Herman|date=December 1987|title=Linograms in Image Reconstruction from Projections|journal=IEEE Transactions on Medical Imaging|volume=MI-6|issue=4|pages=301–7|doi=10.1109/tmi.1987.4307847|pmid=18244038|s2cid=20832295}}&lt;/ref&gt; Once the scan data has been acquired, it is then processed using a form of [[w:tomographic reconstruction|tomographic reconstruction]], which produces a series of cross-sectional images.&lt;ref&gt;{{Cite web|url=https://radiologykey.com/ct-image-reconstruction-basics/|title=CT Image Reconstruction Basics|last=Themes|first=U. F. O.|date=2018-10-07|website=Radiology Key|language=en-US|access-date=2021-03-20}}&lt;/ref&gt; These cross-sectional images are made up of small units of pixels or voxels.&lt;ref name=&quot;Cardiovascular Computed Tomography&quot;&gt;{{Cite book|url=https://books.google.com/books?id=SarDDwAAQBAJ&amp;q=ct+images+are+made+of+pixels&amp;pg=PA134|title=Cardiovascular Computed Tomography|last=Stirrup|first=James|date=2020-01-02|publisher=Oxford University Press|isbn=978-0-19-880927-2|language=en}}&lt;/ref&gt;

A [[w:pixel|pixel]] is a two dimensional unit based on the matrix size and the field of view. [[w:Pixel|Pixel]]s in an image obtained by CT scanning are displayed in terms of relative [[w:radiodensity|radiodensity]]. The pixel itself is displayed according to the mean [[w:attenuation|attenuation]] of the tissue(s) that it corresponds to on a scale from +3,071 (most attenuating) to −1,024 (least attenuating) on the [[w:Hounsfield scale|Hounsfield scale]]. When the CT slice thickness is also factored in, the unit is known as a [[w:voxel|voxel]], which is a three-dimensional unit.&lt;ref name=&quot;Cardiovascular Computed Tomography&quot; /&gt;

Water has an attenuation of 0 [[w:Hounsfield units|Hounsfield units]] (HU), while air is −1,000&amp;nbsp;HU, cancellous bone is typically +400&amp;nbsp;HU, and cranial bone can reach 2,000&amp;nbsp;HU or more (os temporale) and can cause [[w:artifact (error)#Medical imaging|artifacts]]. The attenuation of metallic implants depends on the atomic number of the element used: Titanium usually has an amount of +1000&amp;nbsp;HU, iron steel can completely block the X-ray and is, therefore, responsible for well-known line-artifacts in computed tomograms. Artifacts are caused by abrupt transitions between low- and high-density materials, which results in data values that exceed the dynamic range of the processing electronics. 

Initially, the CT scanners generated images in only [[w:transverse plane|transverse]] (axial) [[w:anatomical plane|anatomical plane]], perpendicular to the long axis of the body. Modern scanners allow the scan data to be reformatted as images in other [[w:Plane (geometry)|planes]]. [[w:Geometry processing|Digital geometry processing]] can generate a [[w:three-dimensional space|three-dimensional]] image of an object inside the body from a series of two-dimensional [[w:radiography|radiographic]] images taken by [[w:rotation around a fixed axis|rotation around a fixed axis]].&lt;ref name=&quot;ref1&quot;&gt;{{Cite book|url=https://books.google.com/books?id=JX__lLLXFHkC&amp;q=ct+can+have+a+number+of+artifacts&amp;pg=PA167|title=Computed Tomography: Principles, Design, Artifacts, and Recent Advances|last=Hsieh|first=Jiang|date=2003|publisher=SPIE Press|isbn=978-0-8194-4425-7|pages=167|language=en}}&lt;/ref&gt; These cross-sectional images are widely used for medical [[w:diagnosis|diagnosis]] and [[w:therapy|therapy]].&lt;ref name=&quot;urlcomputed tomography – Definition from the Merriam-Webster Online Dictionary&quot;&gt;{{Cite web|url=http://www.merriam-webster.com/dictionary/computed+tomography|title=computed tomography – Definition from the Merriam-Webster Online Dictionary|archive-url=https://web.archive.org/web/20110919202302/http://www.merriam-webster.com/dictionary/computed+tomography|archive-date=19 September 2011|access-date=18 August 2009|url-status=live}}&lt;/ref&gt;

== Interpretation of results ==
=== Presentation ===
[[File:CT presentation as thin slice, projection and volume rendering.jpg|thumb|224x224px|&lt;small&gt;Types of presentations of CT scans:&lt;/small&gt;]]

The result of a CT scan is a volume of [[w:voxel|voxel]]s, which may be presented to a human observer by various methods, which broadly fit into the following categories:
*Slices (of varying thickness). Thin slice is generally regarded as planes representing a thickness of less than 3 [[w:Millimetre|mm]].&lt;ref name=&quot;Goldman2008&quot;&gt;{{Cite journal |last=Goldman |first=L. W. |year=2008 |title=Principles of CT: Multislice CT |journal=Journal of Nuclear Medicine Technology |volume=36 |issue=2 |pages=57–68 |doi=10.2967/jnmt.107.044826 |issn=0091-4916 |pmid=18483143 |doi-access=free}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{Cite journal |last1=Reis |first1=Eduardo Pontes |last2=Nascimento |first2=Felipe |last3=Aranha |first3=Mateus |last4=Mainetti Secol |first4=Fernando |last5=Machado |first5=Birajara |last6=Felix |first6=Marcelo |last7=Stein |first7=Anouk |last8=Amaro |first8=Edson |date=29 July 2020 |title=Brain Hemorrhage Extended (BHX): Bounding box extrapolation from thick to thin slice CT images v1.1 |journal=PhysioNet |language=en |volume=101 |issue=23 |pages=215–220 |doi=10.13026/9cft-hg92}}&lt;/ref&gt; Thick slice is generally regarded as planes representing a thickness between 3&amp;nbsp;mm and 5&amp;nbsp;mm.&lt;ref name=&quot;:2&quot; /&gt;&lt;ref&gt;{{Cite journal |last1=Park |first1=S. |last2=Chu |first2=L.C. |last3=Hruban |first3=R.H. |last4=Vogelstein |first4=B. |last5=Kinzler |first5=K.W. |last6=Yuille |first6=A.L. |last7=Fouladi |first7=D.F. |last8=Shayesteh |first8=S. |last9=Ghandili |first9=S. |last10=Wolfgang |first10=C.L. |last11=Burkhart |first11=R. |last12=He |first12=J. |last13=Fishman |first13=E.K. |last14=Kawamoto |first14=S. |date=2020-09-01 |title=Differentiating autoimmune pancreatitis from pancreatic ductal adenocarcinoma with CT radiomics features |journal=Diagnostic and Interventional Imaging |language=en |volume=101 |issue=9 |pages=555–564 |doi=10.1016/j.diii.2020.03.002 |issn=2211-5684 |pmid=32278586 |s2cid=215751181}}&lt;/ref&gt;
*Projection, including [[w:maximum intensity projection|maximum intensity projection]]&lt;ref name=&quot;FishmanNey2006&quot;&gt;{{Cite journal |last1=Fishman |first1=Elliot K. |author-link=Elliot K. Fishman |last2=Ney |first2=Derek R. |last3=Heath |first3=David G. |last4=Corl |first4=Frank M. |last5=Horton |first5=Karen M. |last6=Johnson |first6=Pamela T. |year=2006 |title=Volume Rendering versus Maximum Intensity Projection in CT Angiography: What Works Best, When, and Why |journal=RadioGraphics |volume=26 |issue=3 |pages=905–922 |doi=10.1148/rg.263055186 |issn=0271-5333 |pmid=16702462 |doi-access=free}}&lt;/ref&gt; and ''average intensity projection''
*[[w:Volume rendering|Volume rendering]] (VR)&lt;ref name=&quot;FishmanNey2006&quot; /&gt;

Technically, all volume renderings become projections when viewed on a [[w:Display device#Full-area 2-dimensional displays|2-dimensional display]], making the distinction between projections and volume renderings a bit vague. The epitomes of volume rendering models feature a mix of for example coloring and shading in order to create realistic and observable representations.&lt;ref name=&quot;SilversteinParsad2008&quot;&gt;{{Cite journal |last1=Silverstein |first1=Jonathan C. |last2=Parsad |first2=Nigel M. |last3=Tsirline |first3=Victor |year=2008 |title=Automatic perceptual color map generation for realistic volume visualization |journal=Journal of Biomedical Informatics |volume=41 |issue=6 |pages=927–935 |doi=10.1016/j.jbi.2008.02.008 |issn=1532-0464 |pmc=2651027 |pmid=18430609}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last=Kobbelt |first=Leif |url=https://books.google.com/books?id=zndnSzkfkXwC |title=Vision, Modeling, and Visualization 2006: Proceedings, November 22-24, 2006, Aachen, Germany |date=2006 |publisher=IOS Press |isbn=978-3-89838-081-2 |pages=185 |language=en}}&lt;/ref&gt;

Two-dimensional CT images are conventionally rendered so that the view is as though looking up at it from the patient's feet. Hence, the left side of the image is to the patient's right and vice versa, while anterior in the image also is the patient's anterior and vice versa. This left-right interchange corresponds to the view that physicians generally have in reality when positioned in front of patients.&lt;ref&gt;{{Cite journal |last1=Schmidt |first1=Derek |last2=Odland |first2=Rick |date=September 2004 |title=Mirror-Image Reversal of Coronal Computed Tomography Scans |journal=The Laryngoscope |language=en |volume=114 |issue=9 |pages=1562–1565 |doi=10.1097/00005537-200409000-00011 |issn=0023-852X |pmid=15475782 |s2cid=22320649}}&lt;/ref&gt;

==== Grayscale ====
[[w:Pixel|Pixel]]s in an image obtained by CT scanning are displayed in terms of relative [[w:radiodensity|radiodensity]]. The pixel itself is displayed according to the mean [[w:attenuation|attenuation]] of the tissue(s) that it corresponds to on a scale from +3,071 (most attenuating) to −1,024 (least attenuating) on the [[w:Hounsfield scale|Hounsfield scale]]. A [[w:pixel|pixel]] is a two dimensional unit based on the matrix size and the field of view. When the CT slice thickness is also factored in, the unit is known as a [[w:voxel|voxel]], which is a three-dimensional unit.&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=63xxDwAAQBAJ |title=Brant and Helms' fundamentals of diagnostic radiology |date=2018-07-19 |publisher=Lippincott Williams &amp; Wilkins |isbn=978-1-4963-6738-9 |edition=Fifth |pages=1600 |access-date=24 January 2019}}&lt;/ref&gt; Water has an attenuation of 0 [[w:Hounsfield units|Hounsfield units]] (HU), while air is −1,000&amp;nbsp;HU, cancellous bone is typically +400&amp;nbsp;HU, and cranial bone can reach 2,000&amp;nbsp;HU.&lt;ref&gt;{{Cite book |title=Brain mapping: the methods |date=2002 |publisher=Academic Press |isbn=0-12-693019-8 |editor-last=Arthur W. Toga |edition=2nd |location=Amsterdam |oclc=52594824 |editor-last2=John C. Mazziotta}}&lt;/ref&gt; The attenuation of metallic implants depends on the atomic number of the element used: Titanium usually has an amount of +1000&amp;nbsp;HU, iron steel can completely block the X-ray and is, therefore, responsible for well-known line-artifacts in computed tomograms. Artifacts are caused by abrupt transitions between low- and high-density materials, which results in data values that exceed the dynamic range of the processing electronics.&lt;ref name=&quot;...&quot;&gt;{{Cite book |last1=Jerrold T. Bushberg |title=The essential physics of medical imaging |last2=J. Anthony Seibert |last3=Edwin M. Leidholdt |last4=John M. Boone |date=2002 |publisher=Lippincott Williams &amp; Wilkins |isbn=0-683-30118-7 |edition=2nd |location=Philadelphia |page=358 |oclc=47177732}}&lt;/ref&gt;

==== Windowing ====
CT data sets have a very high [[w:dynamic range|dynamic range]] which must be reduced for display or printing. This is typically done via a process of &quot;windowing&quot;, which maps a range (the &quot;window&quot;) of pixel values to a grayscale ramp. For example, CT images of the brain are commonly viewed with a window extending from 0 HU to 80 HU. Pixel values of 0 and lower, are displayed as black; values of 80 and higher are displayed as white; values within the window are displayed as a grey intensity proportional to position within the window.&lt;ref&gt;{{Cite journal |last1=Kamalian |first1=Shervin |last2=Lev |first2=Michael H. |last3=Gupta |first3=Rajiv |date=2016-01-01 |title=Computed tomography imaging and angiography – principles |journal=Handbook of Clinical Neurology |language=en |volume=135 |pages=3–20 |doi=10.1016/B978-0-444-53485-9.00001-5 |isbn=978-0-444-53485-9 |issn=0072-9752 |pmid=27432657}}&lt;/ref&gt; The window used for display must be matched to the X-ray density of the object of interest, in order to optimize the visible detail.&lt;ref&gt;{{Cite book |last=Stirrup |first=James |url=https://books.google.com/books?id=SarDDwAAQBAJ&amp;q=windowing+in+ct&amp;pg=PA136 |title=Cardiovascular Computed Tomography |date=2020-01-02 |publisher=Oxford University Press |isbn=978-0-19-880927-2 |page=136 |language=en}}&lt;/ref&gt; Window width and window level parameters are used to control the windowing of a scan.&lt;ref&gt;{{Cite book |last=Carroll |first=Quinn B. |url=https://books.google.com/books?id=iTwYI5rzeRMC&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA512&amp;dq=window+width+and+window+level&amp;hl=en |title=Practical Radiographic Imaging |date=2007 |publisher=Charles C Thomas Publisher |isbn=978-0-398-08511-7|page=512 |language=en}}&lt;/ref&gt;

==== Multiplanar reconstruction and projections ====
[[File:Ct-workstation-neck.jpg|thumb|Image showing one volume rendering (VR) and multiplanar view of three thin slices in the [[w:axial plane|axial]] (upper right), [[w:sagittal plane|sagittal]] (lower left), and [[w:coronal plane|coronal plane]]s (lower right)|200x200px]]
Multiplanar reconstruction also known as MPR is the process of converting data from one [[w:anatomical plane|anatomical plane]] (usually [[w:Transverse plane|transverse]]) to other planes. It can be used for thin slices as well as projections. Multiplanar reconstruction is possible as present  CT scanners provide almost [[w:isotropy|isotropic]] resolution.&lt;ref name=&quot;ref3&quot;&gt;{{Cite book |last1=Udupa |first1=Jayaram K. |url=https://books.google.com/books?id=aR6PHYluq4oC&amp;q=3D+Imaging+in+Medicine%2C+2nd+Edition |title=3D Imaging in Medicine, Second Edition |last2=Herman |first2=Gabor T. |date=1999-09-28 |publisher=CRC Press |isbn=978-0-8493-3179-4 |language=en}}&lt;/ref&gt; MPR is used almost in every scan. The spine is frequently examined with it.&lt;ref&gt;{{Cite journal |last1=Krupski |first1=Witold |last2=Kurys-Denis |first2=Ewa |last3=Matuszewski |first3=Łukasz |last4=Plezia |first4=Bogusław |date=2007-06-30 |title=Use of multi-planar reconstruction (MPR) and 3-dimentional &amp;#91;sic&amp;#93; (3D) CT to assess stability criteria in C2 vertebral fractures |url=http://www.jpccr.eu/Use-of-multi-planar-reconstruction-MPR-and-3-dimentional-3D-CT-to-assess-stability,71238,0,2.html |journal=Journal of Pre-Clinical and Clinical Research |language=english |volume=1 |issue=1 |pages=80–83 |issn=1898-2395}}&lt;/ref&gt; An image of the spine in axial plane can only show one vertebral bone at a time and cannot show its relation with other vertebral bones. By reformatting the data in other planes, visualization of the relative position can be achieved in sagittal and coronal plane.&lt;ref&gt;{{Cite journal |last=Tins |first=Bernhard |date=2010-10-21 |title=Technical aspects of CT imaging of the spine |journal=Insights into Imaging |volume=1 |issue=5–6 |pages=349–359 |doi=10.1007/s13244-010-0047-2 |issn=1869-4101 |pmc=3259341 |pmid=22347928}}&lt;/ref&gt; New software allows the reconstruction of data in non-orthogonal (oblique) planes, which help in the visualization of organs which are not in orthogonal planes.&lt;ref&gt;{{Cite book |last1=Wolfson |first1=Nikolaj |url=https://books.google.com/books?id=8Y5FDAAAQBAJ&amp;q=Modern+software+allows+reconstruction+in+non-orthogonal&amp;pg=PA373 |title=Orthopedics in Disasters: Orthopedic Injuries in Natural Disasters and Mass Casualty Events |last2=Lerner |first2=Alexander |last3=Roshal |first3=Leonid |date=2016-05-30 |publisher=Springer |isbn=978-3-662-48950-5 |language=en}}&lt;/ref&gt; It is better suited for visualization of the anatomical structure of the bronchi as they do not lie orthogonal to the direction of the scan.&lt;ref&gt;{{Cite journal |last1=Laroia |first1=Archana T |last2=Thompson |first2=Brad H |last3=Laroia |first3=Sandeep T |last4=van Beek |first4=Edwin JR |date=2010-07-28 |title=Modern imaging of the tracheo-bronchial tree |journal=World Journal of Radiology |volume=2 |issue=7 |pages=237–248 |doi=10.4329/wjr.v2.i7.237 |issn=1949-8470 |pmc=2998855 |pmid=21160663}}&lt;/ref&gt;

Curved-plane reconstruction is performed mainly for the evaluation of vessels. This type of reconstruction helps to straighten the bends in a vessel, thereby helping to visualize a whole vessel in a single image or in multiple images. After a vessel has been &quot;straightened&quot;, measurements such as cross-sectional area and length can be made. This is helpful in preoperative assessment of a surgical procedure.&lt;ref&gt;{{Cite journal |last1=Gong |first1=Jing-Shan |last2=Xu |first2=Jian-Min |date=2004-07-01 |title=Role of curved planar reformations using multidetector spiral CT in diagnosis of pancreatic and peripancreatic diseases |journal=World Journal of Gastroenterology |volume=10 |issue=13 |pages=1943–1947 |doi=10.3748/wjg.v10.i13.1943 |issn=1007-9327 |pmc=4572236 |pmid=15222042}}&lt;/ref&gt;

==== Maximum Intensity Projection (MIP) ====
Maximum Intensity Projection is a scan visualization technique which is used to highlight the highest intensity voxels along a specific projection path. In MIP, each pixel in the final image represents the maximum intensity encountered along a ray traced through the volume data. This method is particularly useful in angiography and vascular imaging, where it enhances the visualization of blood vessels by emphasizing the contrast between vessels and surrounding tissues. MIP projections are valuable for detecting abnormalities and assessing the vascular anatomy with greater clarity.&lt;ref&gt;{{Cite journal|last=Mroz|first=Lukas|last2=König|first2=Andreas|last3=Gröller|first3=Eduard|date=2000-06-01|title=Maximum intensity projection at warp speed|url=https://www.sciencedirect.com/science/article/pii/S0097849300000303|journal=Computers &amp; Graphics|volume=24|issue=3|pages=343–352|doi=10.1016/S0097-8493(00)00030-3|issn=0097-8493}}&lt;/ref&gt;

==== Minimum Intensity Projection (MinIP) ====
Minimum Intensity Projection  highlight the lowest intensity values along a specific projection path. MinIP is particularly beneficial in visualizing structures with low attenuation or density, such as airways in lung imaging. By emphasizing low-intensity features, MinIP can enhance the visibility of subtle details and abnormalities that might be overshadowed in other types of reconstructions. This technique is commonly employed in pulmonary studies to improve the assessment of bronchial structures and airway abnormalities.&lt;ref name=&quot;auto&quot;&gt;{{Cite journal|last=Ghonge|first=NitinP|last2=Chowdhury|first2=Veena|date=2018|title=Minimum-intensity projection images in high-resolution computed tomography lung: Technology update|url=https://journals.lww.com/10.4103/lungindia.lungindia_489_17|journal=Lung India|language=en|volume=35|issue=5|pages=439|doi=10.4103/lungindia.lungindia_489_17|issn=0970-2113|pmc=PMC6120307|pmid=30168468}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Hayabuchi|first=Yasunobu|last2=Inoue|first2=Miki|last3=Watanabe|first3=Noriko|last4=Sakata|first4=Miho|last5=Nabo|first5=Manal Mohamed Helmy|last6=Kagami|first6=Shoji|date=2011-06|title=Minimum-intensity projection of multidetector-row computed tomography for assessment of pulmonary hypertension in children with congenital heart disease|url=https://doi.org/10.1016/j.ijcard.2010.01.008|journal=International Journal of Cardiology|volume=149|issue=2|pages=192–198|doi=10.1016/j.ijcard.2010.01.008|issn=0167-5273}}&lt;/ref&gt;

==== Average Intensity Projection ====
In Average Intensity Projection (AIP), the image is displaying the average attenuation of each voxel within the selected volume.&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-540-89232-8|title=CT of the Acute Abdomen|date=2011|publisher=Springer Berlin Heidelberg|isbn=978-3-540-89231-1|editor-last=Taourel|editor-first=Patrice|series=Medical Radiology|location=Berlin, Heidelberg|language=en|doi=10.1007/978-3-540-89232-8}} Pg. 71&lt;/ref&gt; As the slice thickness increases, the image becomes smoother and more akin to conventional projectional radiography.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=xPwG17yFkzcC&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA213&amp;dq=average+intensity+projection+ct&amp;hl=en|title=CT and MR Angiography: Comprehensive Vascular Assessment|last=Rubin|first=Geoffrey D.|last2=Rofsky|first2=Neil M.|date=2012-10-09|publisher=Lippincott Williams &amp; Wilkins|isbn=978-1-4698-0183-4|language=en}}&lt;/ref&gt; AIP is particularly useful for identifying internal structures of solid organs or the walls of hollow structures, such as intestines. 
{| class=&quot;wikitable&quot;
|+Examples of different algorithms of thickening multiplanar reconstructions&lt;ref&gt;{{Cite journal |last1=Dalrymple |first1=Neal C. |last2=Prasad |first2=Srinivasa R. |last3=Freckleton |first3=Michael W. |last4=Chintapalli |first4=Kedar N. |date=September 2005 |title=Informatics in radiology (infoRAD): introduction to the language of three-dimensional imaging with multidetector CT |journal=Radiographics |volume=25 |issue=5 |pages=1409–1428 |doi=10.1148/rg.255055044 |issn=1527-1323 |pmid=16160120}}&lt;/ref&gt;
!Type of projection
!Schematic illustration
!Examples (10&amp;nbsp;mm slabs)
!Description
!Uses
|-
|Average intensity projection (AIP)
|[[File:Average intensity projection.gif|frameless]]
|[[File:Coronal average intensity projection CT thorax.gif|frameless|118x118px]]
|The average attenuation of each voxel is displayed. The image will get smoother as slice thickness increases. It will look more and more similar to conventional [[w:projectional radiography|projectional radiography]] as slice thickness increases.
|Useful for identifying the internal structures of a solid organ or the walls of hollow structures, such as  intestines.
|-
|[[w:Maximum intensity projection|Maximum intensity projection]] (MIP)
|[[File:Maximum intensity projection.gif|frameless]]
|[[File:Coronal maximum intensity projection CT thorax.gif|frameless|118x118px]]
|The voxel with the highest attenuation is displayed. Therefore, high-attenuating structures such as blood vessels filled with contrast media are enhanced.
|Useful for angiographic studies and identification of pulmonary nodules.
|-
|[[w:Minimum intensity projection|Minimum intensity projection]] (MinIP)
|[[File:Minimum intensity projection.gif|frameless]]
|[[File:Coronal minimum intensity projection CT thorax.gif|frameless|117x117px]]
|The voxel with the lowest attenuation is displayed. Therefore, low-attenuating structures such as air spaces are enhanced.
|Useful for assessing the lung parenchyma.&lt;ref name=&quot;auto&quot;/&gt;&lt;ref&gt;{{Cite book|title=CT of the airways|last=Boiselle|first=Phillip M.|last2=Lynch|first2=David A.|date=2008|publisher=Humana Press|isbn=978-1-59745-139-0|series=Contemporary medical imaging|location=Totowa, NJ}} Page 353&lt;/ref&gt;
|}
====  Volume rendering ====
[[File:Volume rendered CT scan of abdominal and pelvic blood vessels.gif|thumb|191x191px|Volume rendered CT scan of abdominal and pelvic blood vessels.]]
A threshold value of radiodensity is set by the operator (e.g., a level that corresponds to bone). With the help of [[w:edge detection|edge detection]] image processing algorithms a 3D model can be constructed from the initial data and displayed on screen. Various thresholds can be used to get multiple models, each anatomical component such as  muscle, bone and cartilage can be differentiated on the basis of different colours given to them. However, this mode of operation cannot show interior structures.&lt;ref&gt;{{Cite journal |last1=Calhoun |first1=Paul S. |last2=Kuszyk |first2=Brian S. |last3=Heath |first3=David G. |last4=Carley |first4=Jennifer C. |last5=Fishman |first5=Elliot K. |date=1999-05-01 |title=Three-dimensional Volume Rendering of Spiral CT Data: Theory and Method |url=https://pubs.rsna.org/doi/full/10.1148/radiographics.19.3.g99ma14745 |journal=RadioGraphics |volume=19 |issue=3 |pages=745–764 |doi=10.1148/radiographics.19.3.g99ma14745 |issn=0271-5333 |pmid=10336201}}&lt;/ref&gt;

Surface rendering is limited technique as it displays only the surfaces that meet a particular threshold density, and which are towards the viewer. However, In [[w:volume rendering|volume rendering]], transparency, colours and [[w:Phong shading|shading]] are used which makes it easy to present a volume in a single image. For example, Pelvic bones could be displayed as semi-transparent, so that, even viewing at an oblique angle one part of the image does not hide another.&lt;ref&gt;{{Cite journal |last1=van Ooijen |first1=P. M. A. |last2=van Geuns |first2=R. J. M. |last3=Rensing |first3=B. J. W. M. |last4=Bongaerts |first4=A. H. H. |last5=de Feyter |first5=P. J. |last6=Oudkerk |first6=M. |date=January 2003 |title=Noninvasive Coronary Imaging Using Electron Beam CT: Surface Rendering Versus Volume Rendering |url=http://www.ajronline.org/doi/10.2214/ajr.180.1.1800223 |journal=American Journal of Roentgenology |language=en |volume=180 |issue=1 |pages=223–226 |doi=10.2214/ajr.180.1.1800223 |issn=0361-803X |pmid=12490509}}&lt;/ref&gt;

=== Image quality ===
The image quality in computed tomography depends upon the fidelity with which the generated images faithfully represent the attenuation values of X-ray beams as they pass through body tissues, as manifested in the resulting CT image. Image quality encompasses the accurate replication of fine details (Spatial Resolution) and minute discrepancies in attenuation (Contrast Resolution) within the depicted image.

==== Dose versus image quality ====
An important issue within radiology today is how to reduce the radiation dose during CT examinations without compromising the image quality. In general, higher radiation doses result in higher-resolution images,&lt;ref name=&quot;Crowther&quot;&gt;{{Cite journal |last1=R. A. Crowther |last2=D. J. DeRosier |last3=A. Klug |year=1970 |title=The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy |journal=Proc. R. Soc. Lond. A |volume=317 |issue=1530 |pages=319–340 |bibcode=1970RSPSA.317..319C |doi=10.1098/rspa.1970.0119 |s2cid=122980366}}&lt;/ref&gt; while lower doses lead to increased image noise and unsharp images. However, increased dosage raises the adverse side effects, including the risk of [[w:radiation-induced cancer|radiation-induced cancer]] – a four-phase abdominal CT gives the same radiation dose as 300 chest X-rays.&lt;ref&gt;{{Cite journal |last1=Nickoloff |first1=Edward L. |last2=Alderson |first2=Philip O. |date=August 2001 |title=Radiation Exposures to Patients from CT: Reality, Public Perception, and Policy |url=http://www.ajronline.org/doi/10.2214/ajr.177.2.1770285 |journal=American Journal of Roentgenology |language=en |volume=177 |issue=2 |pages=285–287 |doi=10.2214/ajr.177.2.1770285 |issn=0361-803X |pmid=11461846}}&lt;/ref&gt; Several methods that can reduce the exposure to ionizing radiation during a CT scan exist.

# New software technology can significantly reduce the required radiation dose. New [[w:Iterative reconstruction|iterative]] [[w:tomographic reconstruction|tomographic reconstruction]] algorithms (''e.g.'', [[w:SAMV (algorithm)|iterative Sparse Asymptotic Minimum Variance]]) could offer [[w:Super-resolution imaging|super-resolution]] without requiring higher radiation dose.&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=hclVAAAAMAAJ&amp;q=iterative+construction+gives+super+resolution |title=Proceedings |date=1995 |publisher=IEEE |page=10 |isbn=9780780324985 |language=en}}&lt;/ref&gt;
# Individualize the examination and adjust the radiation dose to the body type and body organ examined. Different body types and organs require different amounts of radiation.&lt;ref&gt;{{Cite web |title=Radiation – Effects on organs of the body (somatic effects) |url=https://www.britannica.com/science/radiation |access-date=2021-03-21 |website=Encyclopedia Britannica |language=en}}&lt;/ref&gt;
# Higher resolution is not always suitable, such as detection of small pulmonary masses.&lt;ref&gt;{{Cite journal |last=Simpson G |year=2009 |title=Thoracic computed tomography: principles and practice |journal=Australian Prescriber |volume=32 |issue=4 |page=4 |doi=10.18773/austprescr.2009.049 |doi-access=free}}&lt;/ref&gt;

==== Artifacts ====
Although images produced by CT are generally faithful representations of the scanned volume, the technique is susceptible to a number of [[w:artifact (error)#Medical imaging|artifacts]], such as the following:&lt;ref name=&quot;ref1&quot; /&gt;&lt;ref&gt;{{Cite journal |last1=Bhowmik |first1=Ujjal Kumar |last2=Zafar Iqbal, M. |last3=Adhami, Reza R. |date=28 May 2012 |title=Mitigating motion artifacts in FDK based 3D Cone-beam Brain Imaging System using markers |journal=Central European Journal of Engineering |volume=2 |issue=3 |pages=369–382 |bibcode=2012CEJE....2..369B |doi=10.2478/s13531-012-0011-7 |doi-access=free}}&lt;/ref&gt;
[[File:Aufhaertungsartefakte in der CT durch HTEP 86W - CT axial - 001.jpg|thumb|Metal artifact seen on the right side to a hip prosthesis.]]
Streaks are often seen around materials that block most X-rays, such as metal or bone. Numerous factors contribute to these streaks: under sampling, photon starvation, motion, beam hardening, and [[w:Compton scatter|Compton scatter]]. This type of artifact commonly occurs in the posterior fossa of the brain, or if there are metal implants. The streaks can be reduced using newer reconstruction techniques.&lt;ref name=&quot;P. Jin and C. A. Bouman and K. D. Sauer 2013&quot;&gt;{{Cite journal |last1=P. Jin |last2=C. A. Bouman |last3=K. D. Sauer |year=2013 |title=A Method for Simultaneous Image Reconstruction and Beam Hardening Correction |url=https://engineering.purdue.edu/~bouman/publications/pdf/mic2013.pdf |url-status=dead |journal=IEEE Nuclear Science Symp. &amp; Medical Imaging Conf., Seoul, Korea, 2013 |archive-url=https://web.archive.org/web/20140606234132/https://engineering.purdue.edu/~bouman/publications/pdf/mic2013.pdf |archive-date=2014-06-06 |access-date=2014-04-23}}&lt;/ref&gt; Approaches such as metal artifact reduction (MAR) can also reduce this artifact.&lt;ref&gt;{{Cite journal |vauthors=Boas FE, Fleischmann D |year=2011 |title=Evaluation of Two Iterative Techniques for Reducing Metal Artifacts in Computed Tomography |journal=Radiology |volume=259 |issue=3 |pages=894–902 |doi=10.1148/radiol.11101782 |pmid=21357521}}&lt;/ref&gt;&lt;ref name=&quot;mouton13survey&quot;&gt;{{Cite journal |last1=Mouton, A. |last2=Megherbi, N. |last3=Van Slambrouck, K. |last4=Nuyts, J. |last5=Breckon, T.P. |year=2013 |title=An Experimental Survey of Metal Artefact Reduction in Computed Tomography |url=http://www.durham.ac.uk/toby.breckon/publications/papers/mouton13survey.pdf |journal=Journal of X-Ray Science and Technology |volume=21 |issue=2 |pages=193–226 |doi=10.3233/XST-130372 |pmid=23694911 |hdl=1826/8204}}&lt;/ref&gt; MAR techniques include spectral imaging, where CT images are taken with [[w:photons|photons]] of different energy levels, and then synthesized into [[w:monochromatic|monochromatic]] images with special software such as GSI (Gemstone Spectral Imaging).&lt;ref name=&quot;PessisCampagna2013&quot;&gt;{{Cite journal |last1=Pessis |first1=Eric |last2=Campagna |first2=Raphaël |last3=Sverzut |first3=Jean-Michel |last4=Bach |first4=Fabienne |last5=Rodallec |first5=Mathieu |last6=Guerini |first6=Henri |last7=Feydy |first7=Antoine |last8=Drapé |first8=Jean-Luc |year=2013 |title=Virtual Monochromatic Spectral Imaging with Fast Kilovoltage Switching: Reduction of Metal Artifacts at CT |journal=RadioGraphics |volume=33 |issue=2 |pages=573–583 |doi=10.1148/rg.332125124 |issn=0271-5333 |pmid=23479714 |doi-access=free}}&lt;/ref&gt;

Partial volume effect appears as &quot;blurring&quot; of edges. It is due to the scanner being unable to differentiate between a small amount of high-density material (e.g., bone) and a larger amount of lower density  (e.g., cartilage).&lt;ref&gt;{{Cite journal |last1=González Ballester |first1=Miguel Angel |last2=Zisserman |first2=Andrew P. |last3=Brady |first3=Michael |date=December 2002 |title=Estimation of the partial volume effect in MRI |journal=Medical Image Analysis |volume=6 |issue=4 |pages=389–405 |doi=10.1016/s1361-8415(02)00061-0 |issn=1361-8415 |pmid=12494949}}&lt;/ref&gt; The reconstruction assumes that the X-ray attenuation within each voxel is homogeneous; this may not be the case at sharp edges. This is most commonly seen in the z-direction (craniocaudal direction), due to the conventional use of highly [[w:isotropic|anisotropic]] voxels, which have a much lower out-of-plane resolution, than in-plane resolution. This can be partially overcome by scanning using thinner slices, or an isotropic acquisition on a modern scanner.&lt;ref&gt;{{Cite journal |last1=Goldszal |first1=Alberto F. |last2=Pham |first2=Dzung L. |date=2000-01-01 |title=Volumetric Segmentation |journal=Handbook of Medical Imaging |language=en |pages=185–194 |doi=10.1016/B978-012077790-7/50016-3 |isbn=978-0-12-077790-7}}&lt;/ref&gt;

Ring artifact probably the most common mechanical artifact, the image of one or many &quot;rings&quot; appears within an image. They are usually caused by the variations in the response from individual elements in a two dimensional X-ray detector due to defect or miscalibration.&lt;ref name=&quot;Jha&quot;&gt;{{Cite journal |last=Jha |first=Diwaker |date=2014 |title=Adaptive center determination for effective suppression of ring artifacts in tomography images |journal=Applied Physics Letters |volume=105 |issue=14 |pages=143107 |bibcode=2014ApPhL.105n3107J |doi=10.1063/1.4897441}}&lt;/ref&gt; This phenomenon is more frequently encountered in third-generation CT scanners equipped with solid-state detectors. These artifacts manifest as complete circles in sequential scans, or partial rings in helical CT scans.&lt;ref&gt;{{Cite journal|last=Ramasamy|first=Akilesh|last2=Madhan|first2=Balasubramanian|last3=Krishnan|first3=Balasubramanian|date=2018-08-03|title=Ring artefacts in cranial CT|url=https://casereports.bmj.com/lookup/doi/10.1136/bcr-2018-226097|journal=BMJ Case Reports|language=en|pages=bcr–2018–226097|doi=10.1136/bcr-2018-226097|issn=1757-790X|pmc=PMC6078223|pmid=30076165}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|title=The essential physics of medical imaging|date=2012|publisher=Wolters Kluwer, Lippincott Williams &amp; Wilkins|isbn=978-0-7817-8057-5|editor-last=Bushberg|editor-first=Jerrold T.|edition=3. ed|location=Philadelphia}} p. 373&lt;/ref&gt; Ring artifacts can largely be reduced by intensity normalization, also referred to as flat field correction.&lt;ref name=&quot;vvn15&quot;&gt;{{Cite journal |last1=Van Nieuwenhove |first1=V |last2=De Beenhouwer |first2=J |last3=De Carlo |first3=F |last4=Mancini |first4=L |last5=Marone |first5=F |last6=Sijbers |first6=J |date=2015 |title=Dynamic intensity normalization using eigen flat fields in X-ray imaging |url=http://www.zora.uzh.ch/id/eprint/120683/1/oe-23-21-27975.pdf |journal=Optics Express |volume=23 |issue=21 |pages=27975–27989 |bibcode=2015OExpr..2327975V |doi=10.1364/oe.23.027975 |pmid=26480456 |doi-access=free |hdl=10067/1302930151162165141}}&lt;/ref&gt; Remaining rings can be suppressed by a transformation to polar space, where they become linear stripes.&lt;ref name=&quot;Jha&quot; /&gt; A comparative evaluation of ring artefact reduction on X-ray tomography images showed that the method of Sijbers and Postnov can effectively suppress ring artefacts.&lt;ref name=&quot;jsap&quot;&gt;{{Cite journal |vauthors=Sijbers J, Postnov A |date=2004 |title=Reduction of ring artefacts in high resolution micro-CT reconstructions |journal=Phys Med Biol |volume=49 |issue=14 |pages=N247–53 |doi=10.1088/0031-9155/49/14/N06 |pmid=15357205 |s2cid=12744174}}&lt;/ref&gt;[[File:Bewegungsartefakte im CCT 72W - CT - 001.jpg|thumb|191x191px|Motion Artifact]]Noise appears as grain on the image and is caused by a low signal to noise ratio. This occurs more commonly when a thin slice thickness is used. It can also occur when the power supplied to the X-ray tube is insufficient to penetrate the anatomy.&lt;ref&gt;{{Cite book |last1=Newton |first1=Thomas H. |url=https://books.google.com/books?id=2mxsAAAAMAAJ&amp;q=noise+in+computed+tomography |title=Radiology of the Skull and Brain: Technical aspects of computed tomography |last2=Potts |first2=D. Gordon |date=1971 |publisher=Mosby |isbn=978-0-8016-3662-2 |pages=3941–3950 |language=en}}&lt;/ref&gt;

Windmill artifacts is seen as a streaking appearances which can occur when the detectors intersect the reconstruction plane. This can be reduced with filters or a reduction in pitch.&lt;ref&gt;{{Cite book |last1=Brüning |first1=R. |url=https://books.google.com/books?id=ImOlZNOk25sC&amp;q=windmill+artifact+ct&amp;pg=PA44 |title=Protocols for Multislice CT |last2=Küttner |first2=A. |last3=Flohr |first3=T. |date=2006-01-16 |publisher=Springer Science &amp; Business Media |isbn=978-3-540-27273-1 |language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last=Peh |first=Wilfred C. G. |url=https://books.google.com/books?id=sZswDwAAQBAJ&amp;q=windmill+artifact+ct&amp;pg=PA49 |title=Pitfalls in Musculoskeletal Radiology |date=2017-08-11 |publisher=Springer |isbn=978-3-319-53496-1 |language=en}}&lt;/ref&gt;

Beam hardening artefact can give a &quot;cupped appearance&quot; when grayscale is visualized as height. It occurs because conventional sources, like X-ray tubes emit a polychromatic spectrum. Photons of higher [[w:photon energy|photon energy]] levels are typically attenuated less. Because of this, the mean energy of the spectrum increases when passing the object, often described as getting &quot;harder&quot;. This leads to an effect increasingly underestimating material thickness, if not corrected. Many algorithms exist to correct for this artifact. They can be divided in mono- and multi-material methods.&lt;ref name=&quot;P. Jin and C. A. Bouman and K. D. Sauer 2013&quot; /&gt;&lt;ref&gt;{{Cite journal |vauthors=Van de Casteele E, Van Dyck D, Sijbers J, Raman E |year=2004 |title=A model-based correction method for beam hardening artefacts in X-ray microtomography |journal=Journal of X-ray Science and Technology |volume=12 |issue=1 |pages=43–57 |citeseerx=10.1.1.460.6487}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal |vauthors=Van Gompel G, Van Slambrouck K, Defrise M, Batenburg KJ, Sijbers J, Nuyts J |year=2011 |title=Iterative correction of beam hardening artifacts in CT |journal=Medical Physics |volume=38 |issue=1 |pages=36–49 |bibcode=2011MedPh..38S..36V |citeseerx=10.1.1.464.3547 |doi=10.1118/1.3577758 |pmid=21978116}}&lt;/ref&gt;

Motion artifact refers to unwanted distortions of the images caused by patient motion which can be voluntary on involuntary during the scanning process.

== Advantages ==
CT scan has several advantages over traditional [[w:two-dimensional space|two-dimensional]] medical [[w:radiography|radiography]]. First, CT eliminates the superimposition of images of structures outside the area of interest.&lt;ref&gt;{{Cite book |last1=Mikla |first1=Victor I. |url=https://books.google.com/books?id=Y81JrnVA_5sC&amp;q=ct+scan+removes+superimposition&amp;pg=PA37 |title=Medical Imaging Technology |last2=Mikla |first2=Victor V. |date=2013-08-23 |publisher=Elsevier |isbn=978-0-12-417036-0 |page=37 |language=en}}&lt;/ref&gt; Second, CT scans have greater [[w:image resolution|image resolution]], enabling examination of finer details. CT can distinguish between [[w:tissue (biology)|tissues]] that differ in radiographic [[w:density|density]] by  1% or less.&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=rOppAAAAMAAJ&amp;q=CT+can+distinguish+between+tissue |title=Radiology for the Dental Professional |publisher=Elsevier Mosby |year=2008 |isbn=978-0-323-03071-7 |pages=337}}&lt;/ref&gt; Third, CT scanning enables multiplanar reformatted imaging: scan data can be visualized in the [[w:transverse plane|transverse (or axial)]], [[w:Coronal plane|coronal]], or [[w:Sagittal plane|sagittal]] plane, depending on the diagnostic task.&lt;ref&gt;{{Cite book |last=Pasipoularides |first=Ares |url=https://books.google.com/books?id=eMKqdIvxEmQC&amp;q=ct+scan+enables+multiple+plane+reformatting&amp;pg=PA595 |title=Heart's Vortex: Intracardiac Blood Flow Phenomena |date=November 2009 |publisher=PMPH-USA |isbn=978-1-60795-033-2 |pages=595 |language=en}}&lt;/ref&gt; The improved resolution of CT has permitted the development of new investigations. For example, CT [[w:angiography|angiography]] avoids the invasive insertion of a [[w:catheter|catheter]]. CT scanning can perform a [[w:virtual colonoscopy|virtual colonoscopy]] with greater accuracy and less discomfort for the patient than a traditional [[w:colonoscopy|colonoscopy]].&lt;ref name=&quot;Heiken&quot;&gt;{{Cite journal |last1=Heiken |first1=JP |last2=Peterson CM |last3=Menias CO |date=November 2005 |title=Virtual colonoscopy for colorectal cancer screening: current status: Wednesday 5 October 2005, 14:00–16:00 |journal=Cancer Imaging |publisher=International Cancer Imaging Society |volume=5 |issue=Spec No A |pages=S133–S139 |doi=10.1102/1470-7330.2005.0108 |pmc=1665314 |pmid=16361129}}&lt;/ref&gt;&lt;ref name=&quot;pmid16106357&quot;&gt;{{Cite journal |last1=Bielen DJ |last2=Bosmans HT |last3=De Wever LL |last4=Maes |first4=Frederik |last5=Tejpar |first5=Sabine |last6=Vanbeckevoort |first6=Dirk |last7=Marchal |first7=Guy J.F. |display-authors=3 |name-list-style=vanc |date=September 2005 |title=Clinical validation of high-resolution fast spin-echo MR colonography after colon distention with air |journal=J Magn Reson Imaging |volume=22 |issue=3 |pages=400–5 |doi=10.1002/jmri.20397 |pmid=16106357 |doi-access=free |s2cid=22167728}}&lt;/ref&gt; Virtual colonography is far more accurate than a [[w:barium enema|barium enema]] for detection of tumors and uses a lower radiation dose.&lt;ref&gt;{{Cite web |title=CT Colonography |url=https://www.radiologyinfo.org/en/info.cfm?pg=ct_colo |website=Radiologyinfo.org}}&lt;/ref&gt; 

CT is a moderate-to-high [[w:radiation|radiation]] diagnostic technique. The radiation dose for a particular examination depends on multiple factors: volume scanned, patient build, number and type of scan protocol, and desired resolution and image quality.&lt;ref&gt;{{Cite journal |vauthors=Žabić S, Wang Q, Morton T, Brown KM |date=March 2013 |title=A low dose simulation tool for CT systems with energy integrating detectors |journal=Medical Physics |volume=40 |issue=3 |pages=031102 |bibcode=2013MedPh..40c1102Z |doi=10.1118/1.4789628 |pmid=23464282}}&lt;/ref&gt; Two helical CT scanning parameters, tube current and pitch, can be adjusted easily and have a profound effect on radiation. CT scanning is more accurate than two-dimensional radiographs in evaluating anterior interbody fusion,  although they may still over-read the extent of fusion.&lt;ref&gt;Brian R. Subach M.D., F.A.C.S et al.[http://www.spinemd.com/publications/articles/reliability-and-accuracy-of-fine-cut-computed-tomography-scans-to-determine-the-status-of-anterior-interbody-usions-with-metallic-cages &quot;Reliability and accuracy of fine-cut computed tomography scans to determine the status of anterior interbody fusions with metallic cages&quot;] {{webarchive|url=https://web.archive.org/web/20121208184918/http://www.spinemd.com/publications/articles/reliability-and-accuracy-of-fine-cut-computed-tomography-scans-to-determine-the-status-of-anterior-interbody-usions-with-metallic-cages |date=2012-12-08 }}&lt;/ref&gt;

== Adverse effects ==

=== Contrast reactions ===
The administration of contrast agents carries potential risks, as adverse reactions may occur. While most serious reactions are typically observed after intravascular injection, adverse effects may still manifest after oral or intra-cavitary administration, as some contrast medium molecules may be absorbed into the circulation. Contrast reactions reported in CT scans are classified into three severity levels: Mild, Moderate, and Severe Reactions. 
{| class=&quot;sortable wikitable&quot; style=&quot;float: right; margin-left:15px; text-align:center&quot;
|+Types of Contrast Reactions
!Type
!Severity
!Symptoms may include
!Medical attention
|-
|Mild Reaction
|Low
|nausea, headache, vomiting, flushing, pruritus, and a metallic taste.
|Generally not required, typically resolve on their own.
|-
|Moderate Reactions
|Intermediate
|urticaria, severe vomiting, facial edema, bronchospasm,  laryngeal edema, and vasovagal attacks.
|Treatment approach varies based on the specific symptoms.
|-
|Severe Reaction
|High
|respiratory arrest, cardiac arrest, pulmonary edema, convulsions, and cardiogenic shock.
|Immediate medical attention is essential
|}
In the United States half of CT scans are [[w:contrast CT|contrast CT]]s using intravenously injected [[w:radiocontrast agent|radiocontrast agent]]s.&lt;ref name=&quot;Nam2006&quot; /&gt; The most common reactions from these agents are mild, including nausea, vomiting, and an itching rash. Severe life-threatening reactions may rarely occur.&lt;ref name=&quot;Contrast2005&quot;&gt;{{Cite journal |last=Christiansen C |date=2005-04-15 |title=X-ray contrast media – an overview |journal=Toxicology |volume=209 |issue=2 |pages=185–7 |doi=10.1016/j.tox.2004.12.020 |pmid=15767033}}&lt;/ref&gt; Overall reactions occur in 1 to 3% with [[w:nonionic contrast|nonionic contrast]] and 4 to 12% of people with [[w:ionic contrast|ionic contrast]].&lt;ref name=&quot;Wang2011&quot; /&gt; Skin rashes may appear within a week to 3% of people.&lt;ref name=&quot;Contrast2005&quot; /&gt;

The old [[w:radiocontrast agent|radiocontrast agent]]s caused [[w:anaphylaxis|anaphylaxis]] in 1% of cases while the newer, low-osmolar agents cause reactions in 0.01–0.04% of cases.&lt;ref name=&quot;Contrast2005&quot; /&gt;&lt;ref name=&quot;Drug01&quot;&gt;{{Cite journal |vauthors=Drain KL, Volcheck GW |year=2001 |title=Preventing and managing drug-induced anaphylaxis |journal=Drug Safety |volume=24 |issue=11 |pages=843–53 |doi=10.2165/00002018-200124110-00005 |pmid=11665871 |s2cid=24840296}}&lt;/ref&gt; Death occurs in about 2 to 30 people per 1,000,000 administrations, with newer agents being safer.&lt;ref name=&quot;Wang2011&quot;&gt;{{Cite journal |vauthors=Wang H, Wang HS, Liu ZP |date=October 2011 |title=Agents that induce pseudo-allergic reaction |journal=Drug Discov Ther |volume=5 |issue=5 |pages=211–9 |doi=10.5582/ddt.2011.v5.5.211 |pmid=22466368 |s2cid=19001357}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=bEvnfm7V-LIC&amp;pg=PA187 |title=Anaphylaxis and hypersensitivity reactions |date=2010-12-09 |publisher=Humana Press |isbn=978-1-60327-950-5 |editor-last=Castells |editor-first=Mariana C. |location=New York |page=187}}&lt;/ref&gt; There is a higher risk of mortality in those who are female, elderly or in poor health, usually secondary to either anaphylaxis or [[w:acute kidney injury|acute kidney injury]].&lt;ref name=&quot;Nam2006&quot;&gt;{{Cite journal |vauthors=Namasivayam S, Kalra MK, Torres WE, Small WC |date=Jul 2006 |title=Adverse reactions to intravenous iodinated contrast media: a primer for radiologists |journal=Emergency Radiology |volume=12 |issue=5 |pages=210–5 |doi=10.1007/s10140-006-0488-6 |pmid=16688432 |s2cid=28223134}}&lt;/ref&gt;

The contrast agent may induce [[w:contrast-induced nephropathy|contrast-induced nephropathy]].&lt;ref name=&quot;Contrast2009&quot;&gt;{{Cite journal |vauthors=Hasebroock KM, Serkova NJ |date=April 2009 |title=Toxicity of MRI and CT contrast agents |journal=Expert Opinion on Drug Metabolism &amp; Toxicology |volume=5 |issue=4 |pages=403–16 |doi=10.1517/17425250902873796 |pmid=19368492 |s2cid=72557671}}&lt;/ref&gt; This occurs in 2 to 7% of people who receive these agents, with greater risk in those who have pre-existing [[w:kidney failure|kidney failure]],&lt;ref name=&quot;Contrast2009&quot; /&gt; pre-existing [[w:diabetes mellitus|diabetes]], or reduced intravascular volume. People with mild kidney impairment are usually advised to ensure full hydration for several hours before and after the injection. For moderate kidney failure, the use of  [[w:iodinated contrast|iodinated contrast]] should be avoided; this may mean using an alternative technique instead of CT. Those with severe [[w:kidney failure|kidney failure]] requiring [[w:Kidney dialysis|dialysis]] require less strict precautions, as their kidneys have so little function remaining that any further damage would not be noticeable and the dialysis will remove the contrast agent; it is normally recommended, however, to arrange dialysis as soon as possible following contrast administration to minimize any adverse effects of the contrast.

In addition to the use of intravenous contrast, orally administered contrast agents are frequently used when examining the abdomen.&lt;ref&gt;{{Cite journal |last1=Rawson |first1=James V. |last2=Pelletier |first2=Allen L. |date=2013-09-01 |title=When to Order Contrast-Enhanced CT |url=https://www.aafp.org/afp/2013/0901/p312.html |journal=American Family Physician |volume=88 |issue=5 |pages=312–316 |issn=0002-838X |pmid=24010394}}&lt;/ref&gt; These are frequently the same as the intravenous contrast agents, merely diluted to approximately 10% of the concentration. However, oral alternatives to iodinated contrast exist, such as very dilute (0.5–1% w/v) [[w:barium sulfate|barium sulfate]] suspensions. Dilute barium sulfate has the advantage that it does not cause allergic-type reactions or kidney failure, but cannot be used in patients with suspected bowel perforation or suspected bowel injury, as leakage of barium sulfate from damaged bowel can cause fatal [[w:peritonitis|peritonitis]].&lt;ref&gt;{{Cite book |last1=Thomsen |first1=Henrik S. |url=https://books.google.com/books?id=Bun1CAAAQBAJ&amp;q=intravenous+contrast+in+ct |title=Trends in Contrast Media |last2=Muller |first2=Robert N. |last3=Mattrey |first3=Robert F. |date=2012-12-06 |publisher=Springer Science &amp; Business Media |isbn=978-3-642-59814-2 |language=en}}&lt;/ref&gt;

Side effects from [[w:contrast agent|contrast agent]]s, administered [[w:Intravenous therapy|intravenously]]  in some CT scans, might impair [[w:kidney|kidney]] performance in patients with [[w:kidney disease|kidney disease]], although this risk is now believed to be lower than previously thought.&lt;ref&gt;{{Cite journal |last=Davenport |first=Matthew |year=2020 |title=Use of Intravenous Iodinated Contrast Media in Patients with Kidney Disease: Consensus Statements from the American College of Radiology and the National Kidney Foundation |journal=Radiology |volume=294 |issue=3 |pages=660–668 |doi=10.1148/radiol.2019192094 |pmid=31961246 |doi-access=free}}&lt;/ref&gt;&lt;ref name=&quot;Contrast2009&quot; /&gt;

==== Extravasation ====
Extravasation refers to the unintended leakage of contrast medium, from the intravascular space into the surrounding tissues.&lt;ref&gt;{{Cite journal|last=Alexander|first=Leon|last2=Sheikh Khalifa Medical City, Division of Plastic Surgery, Department of Surgery, Abu Dhabi, UAE|date=2020-09-01|title=Extravasation Injuries: A Trivial Injury Often Overlooked with Disastrous Consequences|url=http://wjps.ir/article-1-637-en.html|journal=WORLD JOURNAL OF PLASTIC SURGERY|language=en|volume=9|issue=3|pages=326–330|doi=10.29252/wjps.9.3.326|issn=2228-7914|pmc=PMC7734938|pmid=33330011}}&lt;/ref&gt; This phenomenon can occur due to various mechanisms during medical procedures involving injection of contrast medium. The fluid can seep into perivascular tissue either by extraluminal dislocation of cannula. Additionally, a leak may manifest at the puncture site of a properly positioned cannula, causing extravasation. The direct impact of pressure from the jet against the vessel wall can lead to vessel disruption and, consequently, extravasation.&lt;ref&gt;{{Cite journal|last=Sakellariou|first=Sophia|last2=Li|first2=Wenguang|last3=Paul|first3=Manosh C|last4=Roditi|first4=Giles|date=2016-12|title=Rôle of contrast media viscosity in altering vessel wall shear stress and relation to the risk of contrast extravasations|url=https://linkinghub.elsevier.com/retrieve/pii/S1350453316302168|journal=Medical Engineering &amp; Physics|language=en|volume=38|issue=12|pages=1426–1433|doi=10.1016/j.medengphy.2016.09.016}}&lt;/ref&gt; Extravasation can arise from both manual hand injection and injections administered using a press-injector. Patients at a greater risk of experiencing extravasation encompass the elderly, infants, children, individuals with impaired consciousness, and those with pre-existing vascular conditions.&lt;ref&gt;{{Cite journal|last=Wang|first=Carolyn L.|last2=Cohan|first2=Richard H.|last3=Ellis|first3=James H.|last4=Adusumilli|first4=Saroja|last5=Dunnick|first5=N. Reed|date=2007-04|title=Frequency, Management, and Outcome of Extravasation of Nonionic Iodinated Contrast Medium in 69 657 Intravenous Injections|url=http://pubs.rsna.org/doi/10.1148/radiol.2431060554|journal=Radiology|language=en|volume=243|issue=1|pages=80–87|doi=10.1148/radiol.2431060554|issn=0033-8419}}&lt;/ref&gt;

Moderate extravasation is characterized by symptoms, such as skin blistering, progressive edema, or ulceration. Close monitoring is advised, and physician assessment is recommended to evaluate for potential neurovascular compromise. This assessment includes checking peripheral pulse and assessing sensation distal to the affected limb. Severe extravasation represents a critical condition with the potential for neurovascular compromise, signs of tissue necrosis, or compartment syndrome. Urgent surgical attention, specifically emergency fasciotomy, is necessary to alleviate pressure within the affected compartment and prevent further complications. &lt;ref&gt;{{Cite journal|last=Liu|first=Wanli|last2=Wang|first2=Pinghu|last3=Zhu|first3=Hui|last4=Tang|first4=Hui|last5=Guan|first5=Hongmei|last6=Wang|first6=Xiaoying|last7=Wang|first7=Chengxiang|last8=Qiu|first8=Yao|last9=He|first9=Lianxiang|date=2023-10-25|title=Contrast media extravasation injury: a prospective observational cohort study|url=https://doi.org/10.1186/s40001-023-01444-5|journal=European Journal of Medical Research|volume=28|issue=1|pages=458|doi=10.1186/s40001-023-01444-5|issn=2047-783X|pmc=PMC10598951|pmid=37880738}}&lt;/ref&gt;

=== Scan dose ===
{| class=&quot;sortable wikitable&quot; style=&quot;float: right; margin-left:15px; text-align:center&quot;
|-
!Examination
!Typical [[w:Effective dose (radiation safety)|effective &lt;br /&gt; dose]] ([[w:Sievert|mSv]])&lt;br /&gt; to the whole body
!Typical [[w:Absorbed dose|absorbed &lt;br /&gt; dose]] ([[w:Gray (unit)|mGy]])&lt;br /&gt; to the organ in question
|-
|Annual background radiation
|2.4&lt;ref name=&quot;background&quot; /&gt;
|2.4&lt;ref name=&quot;background&quot; /&gt;
|-
|Chest X-ray
|0.02&lt;ref name=&quot;FDADose&quot;&gt;{{Cite web |year=2009 |title=What are the Radiation Risks from CT? |url=https://www.fda.gov/radiation-emittingproducts/radiationemittingproductsandprocedures/medicalimaging/medicalX-rays/ucm115329.htm |url-status=live |archive-url=https://web.archive.org/web/20131105050317/https://www.fda.gov/Radiation-EmittingProducts/RadiationEmittingProductsandProcedures/MedicalImaging/MedicalX-Rays/ucm115329.htm |archive-date=2013-11-05 |website=Food and Drug Administration}}&lt;/ref&gt;
|0.01–0.15&lt;ref name=&quot;crfdr&quot; /&gt;
|-
|Head CT
|1–2&lt;ref name=&quot;Furlow2010&quot; /&gt;
|56&lt;ref name=&quot;nrpb2005&quot;&gt;Shrimpton, P.C; Miller, H.C; Lewis, M.A; Dunn, M.  [http://www.hpa.org.uk/web/HPAwebFile/HPAweb_C/1194947420292 Doses from Computed Tomography (CT) examinations in the UK – 2003 Review] {{webarchive|url=https://web.archive.org/web/20110922122151/http://www.hpa.org.uk/web/HPAwebFile/HPAweb_C/1194947420292 |date=2011-09-22 }}&lt;/ref&gt;
|-
|Screening [[mammography]]
|0.4&lt;ref name=&quot;Risk2011&quot;&gt;{{Cite journal |last1=Davies |first1=H. E. |last2=Wathen, C. G. |last3=Gleeson, F. V. |date=25 February 2011 |title=The risks of radiation exposure related to diagnostic imaging and how to minimise them |journal=BMJ |volume=342 |issue=feb25 1 |pages=d947 |doi=10.1136/bmj.d947 |pmid=21355025 |s2cid=206894472}}&lt;/ref&gt;
|3&lt;ref name=&quot;Brenner2007&quot;&gt;{{Cite journal|date=November 2007|title=Computed tomography – an increasing source of radiation exposure|url=http://www.columbia.edu/~djb3/papers/nejm1.pdf|journal=N. Engl. J. Med.|volume=357|issue=22|pages=2277–84|doi=10.1056/NEJMra072149|pmid=18046031|archive-url=https://web.archive.org/web/20160304060542/http://www.columbia.edu/~djb3/papers/nejm1.pdf|archive-date=2016-03-04|vauthors=Brenner DJ, Hall EJ|url-status=live|s2cid=2760372}}&lt;/ref&gt;&lt;ref name=&quot;crfdr&quot; /&gt;
|-
|Abdominal CT
|8&lt;ref name=&quot;FDADose&quot; /&gt;
|14&lt;ref name=&quot;nrpb2005&quot; /&gt;
|-
|Chest CT
|5–7&lt;ref name=&quot;Furlow2010&quot; /&gt;
|13&lt;ref name=&quot;nrpb2005&quot; /&gt;
|-
|[[Virtual colonoscopy|CT colonography]]
|6–11&lt;ref name=&quot;Furlow2010&quot; /&gt;
|
|-
|Chest, abdomen and pelvis CT
|9.9&lt;ref name=&quot;nrpb2005&quot; /&gt;
|12&lt;ref name=&quot;nrpb2005&quot; /&gt;
|-
|Cardiac CT angiogram
|9–12&lt;ref name=&quot;Furlow2010&quot; /&gt;
|40–100&lt;ref name=&quot;crfdr&quot; /&gt;
|-
|[[Barium enema]]
|15&lt;ref name=&quot;Brenner2007&quot; /&gt;
|15&lt;ref name=&quot;crfdr&quot; /&gt;
|-
|Neonatal abdominal CT
|20&lt;ref name=&quot;Brenner2007&quot; /&gt;
|20&lt;ref name=&quot;crfdr&quot; /&gt;
|-
| colspan=&quot;3&quot; |
|}
The magnitude of radiation exposure encountered by patients undergoing computed tomography examinations depends on the scanner's design,&lt;ref&gt;{{Cite journal|last=Jaffe|first=Tracy A.|last2=Yoshizumi|first2=Terry T.|last3=Toncheva|first3=Greta|last4=Anderson-Evans|first4=Colin|last5=Lowry|first5=Carolyn|last6=Miller|first6=Chad M.|last7=Nelson|first7=Rendon C.|last8=Ravin|first8=Carl E.|date=2009-10|title=Radiation Dose for Body CT Protocols: Variability of Scanners at One Institution|url=https://www.ajronline.org/doi/10.2214/AJR.09.2330|journal=American Journal of Roentgenology|language=en|volume=193|issue=4|pages=1141–1147|doi=10.2214/AJR.09.2330|issn=0361-803X}}&lt;/ref&gt; and the factors chosen by the radiology technologist including current, voltage, scan field, scan duration, filtration, rotation angle, collimation, and section thickness and spacing, collectively contribute to the overall cumulative dose.&lt;ref&gt;{{Cite journal|last=Rothenberg|first=L N|last2=Pentlow|first2=K S|date=1992-11|title=Radiation dose in CT.|url=http://pubs.rsna.org/doi/10.1148/radiographics.12.6.1439023|journal=RadioGraphics|language=en|volume=12|issue=6|pages=1225–1243|doi=10.1148/radiographics.12.6.1439023|issn=0271-5333}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Kalra|first=Mannudeep K.|last2=Maher|first2=Michael M.|last3=Toth|first3=Thomas L.|last4=Hamberg|first4=Leena M.|last5=Blake|first5=Michael A.|last6=Shepard|first6=Jo-Anne|last7=Saini|first7=Sanjay|date=2004-03|title=Strategies for CT Radiation Dose Optimization|url=http://pubs.rsna.org/doi/10.1148/radiol.2303021726|journal=Radiology|language=en|volume=230|issue=3|pages=619–628|doi=10.1148/radiol.2303021726|issn=0033-8419}}&lt;/ref&gt; These considerations are important for optimizing both equipment configurations and scanning protocols to have a balance between achieving diagnostic precision and minimizing radiation exposure to patients.&lt;ref&gt;{{Cite journal|last=Toth|first=Thomas L.|date=2002-04-01|title=Dose reduction opportunities for CT scanners|url=https://doi.org/10.1007/s00247-002-0678-7|journal=Pediatric Radiology|language=en|volume=32|issue=4|pages=261–267|doi=10.1007/s00247-002-0678-7|issn=1432-1998}}&lt;/ref&gt;

The table reports average radiation exposures; however, there can be a wide variation in radiation doses between similar scan types, where the highest dose could be as much as 22 times higher than the lowest dose.&lt;ref name=&quot;Furlow2010&quot; /&gt;  A typical plain film X-ray involves radiation dose of 0.01 to 0.15&amp;nbsp;mGy, while a typical CT can involve 10–20&amp;nbsp;mGy for specific organs, and can go up to 80&amp;nbsp;mGy for certain specialized CT scans.&lt;ref name=&quot;crfdr&quot;&gt;{{Cite journal |vauthors=Hall EJ, Brenner DJ |date=May 2008 |title=Cancer risks from diagnostic radiology |journal=The British Journal of Radiology |volume=81 |issue=965 |pages=362–78 |doi=10.1259/bjr/01948454 |pmid=18440940 |s2cid=23348032}}&lt;/ref&gt;

For purposes of comparison, the world average dose rate from naturally occurring sources of [[w:background radiation|background radiation]] is 2.4&amp;nbsp;[[w:mSv|mSv]] per year, equal for practical purposes in this application to 2.4&amp;nbsp;mGy per year.&lt;ref name=&quot;background&quot;&gt;{{Cite journal |vauthors=Cuttler JM, Pollycove M |year=2009 |title=Nuclear energy and health: and the benefits of low-dose radiation hormesis |journal=Dose-Response |volume=7 |issue=1 |pages=52–89 |doi=10.2203/dose-response.08-024.Cuttler |pmc=2664640 |pmid=19343116}}&lt;/ref&gt; While there is some variation, most people (99%) received less than 7&amp;nbsp;mSv per year as background radiation.&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=qCebxPjdSBUC&amp;pg=PA164 |title=A half century of health physics |publisher=Lippincott Williams &amp; Wilkins |year=2005 |isbn=978-0-7817-6934-1 |editor-last=Michael T. Ryan |location=Baltimore, Md. |page=164 |editor-last2=Poston, John W.}}&lt;/ref&gt; Medical imaging as of 2007 accounted for half of the radiation exposure of those in the United States with CT scans making up two thirds of this amount.&lt;ref name=&quot;Furlow2010&quot; /&gt; In the United Kingdom it accounts for 15% of radiation exposure.&lt;ref name=&quot;Risk2011&quot; /&gt; The average radiation dose from medical sources is ≈0.6&amp;nbsp;mSv per person globally as of 2007.&lt;ref name=&quot;Furlow2010&quot; /&gt; Those in the nuclear industry in the United States are limited to doses of 50&amp;nbsp;mSv a year and 100&amp;nbsp;mSv every 5 years.&lt;ref name=&quot;Furlow2010&quot; /&gt;

==== Radiation dose units ====
The radiation dose reported in the [[w:Gray (unit)|gray or mGy]] unit is proportional to the amount of energy that the irradiated body part is expected to absorb, and the physical effect (such as DNA [[w:double strand breaks|double strand breaks]]) on the cells' chemical bonds by X-ray radiation is proportional to that energy.&lt;ref&gt;{{Cite journal |vauthors=Polo SE, Jackson SP |date=March 2011 |title=Dynamics of DNA damage response proteins at DNA breaks: a focus on protein modifications |journal=Genes Dev. |volume=25 |issue=5 |pages=409–33 |doi=10.1101/gad.2021311 |pmc=3049283 |pmid=21363960}}&lt;/ref&gt;

The [[w:sievert|sievert]] unit is used in the report of the [[w:effective dose (radiation)|effective dose]]. The sievert unit, in the context of CT scans, does not correspond to the actual radiation dose that the scanned body part absorbs but to another radiation dose of another scenario, the whole body absorbing the other radiation dose and the other radiation dose being of a magnitude, estimated to have the same probability to induce cancer as the CT scan.&lt;ref&gt;{{Cite journal|last=McCollough|first=Cynthia|last2=Cody|first2=Dianna|last3=Edyvean|first3=Sue|last4=Geise|first4=Rich|last5=Gould|first5=Bob|last6=Keat|first6=Nicholas|last7=Huda|first7=Walter|last8=Judy|first8=Phil|last9=Kalender|first9=Willi|date=2008-01|title=The Measurement, Reporting, and Management of Radiation Dose in CT|url=https://doi.org/10.37206/97|doi=10.37206/97}}&lt;/ref&gt; Thus, as is shown in the table above, the actual radiation that is absorbed by a scanned body part is often much larger than the effective dose suggests. A specific measure, termed the [[w:computed tomography dose index|computed tomography dose index]] (CTDI), is commonly used as an estimate of the radiation absorbed dose for tissue within the scan region, and is automatically computed by medical CT scanners.&lt;ref&gt;{{Cite journal |vauthors=Hill B, Venning AJ, Baldock C |year=2005 |title=A preliminary study of the novel application of normoxic polymer gel dosimeters for the measurement of CTDI on diagnostic X-ray CT scanners |journal=Medical Physics |volume=32 |issue=6 |pages=1589–1597 |bibcode=2005MedPh..32.1589H |doi=10.1118/1.1925181 |pmid=16013718}}&lt;/ref&gt; It is usually expressed in units of milligray (mGy). CTDI is further divided into CTDI&lt;sub&gt;100&lt;/sub&gt;, CTDI&lt;sub&gt;w&lt;/sub&gt; and CTDI&lt;sub&gt;vol&lt;/sub&gt;. CTDI is expressed mathematically as:&lt;ref&gt;{{Cite journal|last=Leon|first=Stephanie M.|last2=Kobistek|first2=Robert J.|last3=Olguin|first3=Edmond A.|last4=Zhang|first4=Zhongwei|last5=Barreto|first5=Izabella L.|last6=Schwarz|first6=Bryan C.|date=2020-08|title=The helically‐acquired CTDI vol as an alternative to traditional methodology|url=https://aapm.onlinelibrary.wiley.com/doi/10.1002/acm2.12944|journal=Journal of Applied Clinical Medical Physics|language=en|volume=21|issue=8|pages=263–271|doi=10.1002/acm2.12944|issn=1526-9914|pmc=PMC7484853|pmid=32519415}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Mekonin|first=Tadelech S.|last2=Deressu|first2=Tilahun T.|date=2022-07|title=Computed Dosimeter Dose Index on a 16-Slice Computed Tomography Scanner|url=http://journals.sagepub.com/doi/10.1177/15593258221119299|journal=Dose-Response|language=en|volume=20|issue=3|pages=155932582211192|doi=10.1177/15593258221119299|issn=1559-3258|pmc=PMC9403463|pmid=36034103}}&lt;/ref&gt;
{| class=&quot;sortable wikitable&quot; style=&quot;float: right; margin-left:15px; text-align:center&quot;
|+Computed Tomography Dose Index
!Radiation exposure Index
!Formula
!
|-
|Computed tomography dose index
|&lt;math display=&quot;block&quot;&gt;CTDI=\frac{1}{nT}\int_{-z}^{+z}{D(z)\text{d}z}&lt;/math&gt;
|Where &lt;math&gt;n&lt;/math&gt; represent the number of slices obtained during a single axial rotation, &lt;math&gt;T&lt;/math&gt; denote the width of an individual acquired slice, and &lt;math&gt;D(z)&lt;/math&gt; represent the radiation dose recorded at position &lt;math&gt;z&lt;/math&gt; along the primary axis of the scanner.&lt;ref name=&quot;auto3&quot;&gt;{{Cite journal|last=Teeuwisse|first=W M|last2=Geleijns|first2=J|last3=Broerse|first3=J J|last4=Obermann|first4=W R|last5=van Persijn van Meerten|first5=E L|date=2001-08|title=Patient and staff dose during CT guided biopsy, drainage and coagulation|url=http://www.birpublications.org/doi/10.1259/bjr.74.884.740720|journal=The British Journal of Radiology|language=en|volume=74|issue=884|pages=720–726|doi=10.1259/bjr.74.884.740720|issn=0007-1285}}&lt;/ref&gt;
|-
|Computed tomography dose index 100
|&lt;math&gt;CTDI_{100}=\frac{1}{nT}\int_{-50 mm}^{50 mm}{D(z)\text{d}z}&lt;/math&gt;.
|
|-
|Computed tomography dose index weighted
|&lt;math&gt;CTDI_w=\frac{1}{3} CTDI_{100}^{central} + \frac{2}{3} CTDI_{100}^{peripheral}&lt;/math&gt;
|&lt;math&gt;CTDI^{central}&lt;/math&gt; is measured at center and &lt;math&gt;CTDI^{peripheral}&lt;/math&gt; is measured at the periphery.&lt;ref name=&quot;auto3&quot;/&gt;
|-
|Computed tomography dose index volume
|&lt;math&gt;CTDI_{vol}=\frac{CTDI_{w}}{P}&lt;/math&gt;
|
|}

Another important parameter in assessing radiation dose in CT is the Dose Length Product (DLP) which is calculated as the product of CTDI&lt;sub&gt;vol&lt;/sub&gt; and the scan length expressed as mGy*cm. It provides an indication of the overall dose output, taking into consideration the length of the scan. However, DLP does not account for the size of the patient and is not a direct measure of absorbed dose or effective dose.To account for variations in patient size, an additional metric, Size-Specific Dose Estimate (SSDE), is introduced. SSDE, measured in mGy, takes into account the size of the patient, providing a more estimate of absorbed dose. It focuses on accounting for variations in patient size when estimating the absorbed dose during a CT examination

The [[w:equivalent dose|equivalent dose]] is the effective dose of a case, in which the whole body would actually absorb the same radiation dose, and the sievert unit is used in its report. In the case of non-uniform radiation, or radiation given to only part of the body, which is common for CT examinations, using the local equivalent dose alone would overstate the biological risks to the entire organism.&lt;ref&gt;{{Cite book |last1=Issa |first1=Ziad F. |title=Clinical Arrhythmology and Electrophysiology |last2=Miller |first2=John M. |last3=Zipes |first3=Douglas P. |date=2019-01-01 |publisher=Elsevier |isbn=978-0-323-52356-1 |pages=1042–1067 |language=en |chapter=Complications of Catheter Ablation of Cardiac Arrhythmias |doi=10.1016/b978-0-323-52356-1.00032-3}}&lt;/ref&gt;&lt;ref&gt;{{Cite web |title=Absorbed, Equivalent, and Effective Dose – ICRPaedia |url=http://icrpaedia.org/Absorbed,_Equivalent,_and_Effective_Dose |access-date=2021-03-21 |website=icrpaedia.org}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |last=Materials |first=National Research Council (US) Committee on Evaluation of EPA Guidelines for Exposure to Naturally Occurring Radioactive |url=https://www.ncbi.nlm.nih.gov/books/NBK230653/ |title=Radiation Quantities and Units, Definitions, Acronyms |date=1999 |publisher=National Academies Press (US) |language=en}}&lt;/ref&gt;

==== Effects of radiation ====
Most adverse health effects of radiation exposure may be grouped in two general categories:
*deterministic effects (harmful tissue reactions) due in large part to the killing/malfunction of cells following high doses;&lt;ref&gt;{{Cite book |last1=Pua |first1=Bradley B. |url=https://books.google.com/books?id=7fpyDwAAQBAJ&amp;q=deterministic+effects&amp;pg=PA53 |title=Interventional Radiology: Fundamentals of Clinical Practice |last2=Covey |first2=Anne M. |last3=Madoff |first3=David C. |date=2018-12-03 |publisher=Oxford University Press |isbn=978-0-19-027624-9 |language=en}}&lt;/ref&gt;
*stochastic effects, i.e., cancer and heritable effects involving either cancer development in exposed individuals owing to mutation of somatic cells or heritable disease in their offspring owing to mutation of reproductive (germ) cells.&lt;ref&gt;Paragraph 55 in: {{Cite web |title=The 2007 Recommendations of the International Commission on Radiological Protection |url=http://www.icrp.org/publication.asp?id=ICRP%20Publication%20103 |url-status=live |archive-url=https://web.archive.org/web/20121116084754/http://www.icrp.org/publication.asp?id=ICRP+Publication+103 |archive-date=2012-11-16 |website=[[International Commission on Radiological Protection]]}} Ann. ICRP 37 (2-4)&lt;/ref&gt;

The added lifetime risk of developing cancer by a single abdominal CT of 8 mSv is estimated to be 0.05%, or 1 one in 2,000.&lt;ref&gt;{{Cite web |date=March 2013 |title=Do CT scans cause cancer? |url=https://www.health.harvard.edu/staying-healthy/do-ct-scans-cause-cancer |url-status=dead |archive-url=https://web.archive.org/web/20171209152338/https://www.health.harvard.edu/staying-healthy/do-ct-scans-cause-cancer |archive-date=2017-12-09 |access-date=2017-12-09 |website=[[Harvard Medical School]]}}&lt;/ref&gt;

Because of increased susceptibility of fetuses to radiation exposure, the radiation dosage of a CT scan is an important consideration in the choice of [[w:medical imaging in pregnancy|medical imaging in pregnancy]].&lt;ref&gt;{{Cite web |last=CDC |date=2020-06-05 |title=Radiation and pregnancy: A fact sheet for clinicians |url=https://www.cdc.gov/nceh/radiation/emergencies/prenatalphysician.htm |access-date=2021-03-21 |website=Centers for Disease Control and Prevention |language=en-us}}&lt;/ref&gt;&lt;ref&gt;{{Citation |last1=Yoon |first1=Ilsup |title=Radiation Exposure In Pregnancy |date=2021 |url=http://www.ncbi.nlm.nih.gov/books/NBK551690/ |work=StatPearls |place=Treasure Island (FL) |publisher=StatPearls Publishing |pmid=31869154 |access-date=2021-03-21 |last2=Slesinger |first2=Todd L.}}&lt;/ref&gt;

==== Excess doses ====
In October, 2009, the US [[w:Food and Drug Administration|Food and Drug Administration]] (FDA) initiated an investigation of brain perfusion CT (PCT) scans, based on [[w:radiation burn|radiation burn]]s caused by incorrect settings at one particular facility for this particular type of CT scan. Over 256 patients were exposed to radiations for over 18-month period. Over 40% of them lost patches of hair, and prompted the editorial to call for increased CT quality assurance programs. It was noted that &quot;while unnecessary radiation exposure should be avoided, a medically needed CT scan obtained with appropriate acquisition parameter has benefits that outweigh the radiation risks.&quot;&lt;ref name=&quot;Furlow2010&quot;&gt;{{Cite book |last=Whaites |first=Eric |url=https://books.google.com/books?id=qdOSDdETuxcC&amp;q=Typical+effective+dose&amp;pg=PA27 |title=Radiography and Radiology for Dental Care Professionals E-Book |date=2008-10-10 |publisher=Elsevier Health Sciences |isbn=978-0-7020-4799-2 |pages=25 |language=en}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal |vauthors=Wintermark M, Lev MH |date=January 2010 |title=FDA investigates the safety of brain perfusion CT |journal=AJNR Am J Neuroradiol |volume=31 |issue=1 |pages=2–3 |doi=10.3174/ajnr.A1967 |pmc=7964089 |pmid=19892810 |doi-access=free}}&lt;/ref&gt; Similar problems have been reported at other centers.&lt;ref name=&quot;Furlow2010&quot; /&gt; These incidents are believed to be due to [[w:human error|human error]].&lt;ref name=&quot;Furlow2010&quot; /&gt;

=== Cancer ===
The [[w:ionizing radiation|radiation]] used in CT scans can damage body cells, including [[w:DNA molecule|DNA molecule]]s, which can lead to [[w:radiation-induced cancer|radiation-induced cancer]].&lt;ref name=&quot;Brenner2007&quot; /&gt; The radiation doses received from CT scans is variable.  Compared to the lowest dose x-ray techniques, CT scans can have 100 to 1,000 times higher dose than conventional X-rays.&lt;ref name=&quot;Redberg&quot;&gt;Redberg, Rita F., and Smith-Bindman, Rebecca. [https://www.nytimes.com/2014/01/31/opinion/we-are-giving-ourselves-cancer.html &quot;We Are Giving Ourselves Cancer&quot;] {{webarchive|url=https://web.archive.org/web/20170706163542/https://www.nytimes.com/2014/01/31/opinion/we-are-giving-ourselves-cancer.html?nl=opinion&amp;emc=edit_ty_20140131&amp;_r=0|date=2017-07-06}}, ''New York Times'', January 30, 2014&lt;/ref&gt; However, a lumbar spine x-ray has a similar dose as a head CT.&lt;ref&gt;{{Cite web|url=https://www.fda.gov/Radiation-EmittingProducts/RadiationEmittingProductsandProcedures/MedicalImaging/MedicalX-Rays/ucm115329.htm|title=Medical X-ray Imaging – What are the Radiation Risks from CT?|last=Health|first=Center for Devices and Radiological|website=www.fda.gov|archive-url=https://web.archive.org/web/20131105050317/https://www.fda.gov/Radiation-EmittingProducts/RadiationEmittingProductsandProcedures/MedicalImaging/MedicalX-Rays/ucm115329.htm|archive-date=5 November 2013|access-date=1 May 2018|url-status=live}}&lt;/ref&gt; Articles in the media often exaggerate the relative dose of CT by comparing the lowest-dose x-ray techniques (chest x-ray) with the highest-dose CT techniques. In general, a routine abdominal CT has a radiation dose similar to three years of average [[w:background radiation|background radiation]].&lt;ref&gt;{{Cite web|url=https://www.acr.org/-/media/ACR/Files/Radiology-Safety/Radiation-Safety/Dose-Reference-Card.pdf|title=Patient Safety – Radiation Dose in X-Ray and CT Exams|last=(ACR)|first=[[Radiological Society of North America]] (RSNA) and [[American College of Radiology]]|date=February 2021|website=acr.org|archive-url=https://web.archive.org/web/20210101161039/https://www.acr.org/-/media/ACR/Files/Radiology-Safety/Radiation-Safety/Dose-Reference-Card.pdf|archive-date=1 January 2021|access-date=6 April 2021|url-status=dead}}&lt;/ref&gt;

Studies published in 2020 on 2.5 million patients&lt;ref name=&quot;Patients undergoing recurrent CT sc&quot;&gt;{{Cite journal|last1=Rehani|first1=Madan M.|last2=Yang|first2=Kai|last3=Melick|first3=Emily R.|last4=Heil|first4=John|last5=Šalát|first5=Dušan|last6=Sensakovic|first6=William F.|last7=Liu|first7=Bob|year=2020|title=Patients undergoing recurrent CT scans: assessing the magnitude|journal=European Radiology|volume=30|issue=4|pages=1828–1836|doi=10.1007/s00330-019-06523-y|pmid=31792585|s2cid=208520824}}&lt;/ref&gt; and 3.2 million patients&lt;ref name=&quot;Multinational data on cumulative ra&quot;&gt;{{Cite journal|last1=Brambilla|first1=Marco|last2=Vassileva|first2=Jenia|last3=Kuchcinska|first3=Agnieszka|last4=Rehani|first4=Madan M.|year=2020|title=Multinational data on cumulative radiation exposure of patients from recurrent radiological procedures: call for action|journal=European Radiology|volume=30|issue=5|pages=2493–2501|doi=10.1007/s00330-019-06528-7|pmid=31792583|s2cid=208520544}}&lt;/ref&gt; have drawn attention to high cumulative doses of more than 100 mSv to patients undergoing recurrent CT scans within a short time span of 1 to 5 years.

Some experts note that CT scans are known to be &quot;overused,&quot; and &quot;there is distressingly little evidence of better health outcomes associated with the current high rate of scans.&quot;&lt;ref name=&quot;Redberg&quot; /&gt; On the other hand, a recent paper analyzing the data of patients who received high [[w:cumulative dose|cumulative dose]]s showed a high degree of appropriate use.&lt;ref name=&quot;Patients undergoing recurrent CT ex&quot;&gt;{{Cite journal|last1=Rehani|first1=Madan M.|last2=Melick|first2=Emily R.|last3=Alvi|first3=Raza M.|last4=Doda Khera|first4=Ruhani|last5=Batool-Anwar|first5=Salma|last6=Neilan|first6=Tomas G.|last7=Bettmann|first7=Michael|year=2020|title=Patients undergoing recurrent CT exams: assessment of patients with non-malignant diseases, reasons for imaging and imaging appropriateness|journal=European Radiology|volume=30|issue=4|pages=1839–1846|doi=10.1007/s00330-019-06551-8|pmid=31792584|s2cid=208520463}}&lt;/ref&gt; This creates an important issue of cancer risk to these patients. Moreover, a highly significant finding that was previously unreported is that some patients received &gt;100 mSv dose from CT scans in  a single day,&lt;ref name=&quot;Patients undergoing recurrent CT sc&quot; /&gt; which counteracts existing criticisms some investigators may have on the effects of protracted versus acute exposure.

Early estimates of harm from CT are partly based on similar radiation exposures experienced by those present during the [[w:atomic bomb|atomic bomb]] explosions in Japan after the [[w:World War II|Second World War]] and those of [[w:nuclear industry|nuclear industry]] workers.&lt;ref name=&quot;Brenner2007&quot; /&gt; Some experts project that in the future, between three and five percent of all cancers would result from medical imaging.&lt;ref name=&quot;Redberg&quot; /&gt;

An Australian study of 10.9&amp;nbsp;million people reported that the increased incidence of cancer after CT scan exposure in this cohort was mostly due to irradiation. In this group, one in every 1,800 CT scans was followed by an excess cancer. If the lifetime risk of developing cancer is 40% then the absolute risk rises to 40.05% after a CT.&lt;ref name=&quot;MathewsForsythe2013&quot;&gt;{{Cite journal|last1=Mathews|first1=J. D.|last2=Forsythe|first2=A. V.|last3=Brady|first3=Z.|last4=Butler|first4=M. W.|last5=Goergen|first5=S. K.|last6=Byrnes|first6=G. B.|last7=Giles|first7=G. G.|last8=Wallace|first8=A. B.|last9=Anderson|first9=P. R.|year=2013|title=Cancer risk in 680 000 people exposed to computed tomography scans in childhood or adolescence: data linkage study of 11 million Australians|journal=BMJ|volume=346|issue=may21 1|pages=f2360|doi=10.1136/bmj.f2360|issn=1756-1833|pmc=3660619|pmid=23694687|last10=Guiver|first10=T. A.|last11=McGale|first11=P.|last12=Cain|first12=T. M.|last13=Dowty|first13=J. G.|last14=Bickerstaffe|first14=A. C.|last15=Darby|first15=S. C.}}&lt;/ref&gt;&lt;ref name=&quot;SasieniShelton2011&quot;&gt;{{Cite journal|last1=Sasieni|first1=P D|last2=Shelton|first2=J|last3=Ormiston-Smith|first3=N|last4=Thomson|first4=C S|last5=Silcocks|first5=P B|year=2011|title=What is the lifetime risk of developing cancer?: the effect of adjusting for multiple primaries|journal=British Journal of Cancer|volume=105|issue=3|pages=460–465|doi=10.1038/bjc.2011.250|issn=0007-0920|pmc=3172907|pmid=21772332}}&lt;/ref&gt;

Some studies have shown that publications indicating an increased risk of cancer from typical doses of body CT scans are plagued with serious methodological limitations and several highly improbable results,&lt;ref&gt;{{Cite journal|last1=Eckel|first1=Laurence J.|last2=Fletcher|first2=Joel G.|last3=Bushberg|first3=Jerrold T.|last4=McCollough|first4=Cynthia H.|date=2015-10-01|title=Answers to Common Questions About the Use and Safety of CT Scans|url=https://www.mayoclinicproceedings.org/article/S0025-6196(15)00591-1/fulltext|journal=Mayo Clinic Proceedings|language=en|volume=90|issue=10|pages=1380–1392|doi=10.1016/j.mayocp.2015.07.011|issn=0025-6196|pmid=26434964|doi-access=free}}&lt;/ref&gt; concluding that no evidence indicates such low doses cause any long-term harm.&lt;ref&gt;{{Cite web|url=https://www.sciencedaily.com/releases/2015/10/151005151507.htm|title=Expert opinion: Are CT scans safe?|website=ScienceDaily|language=en|access-date=2019-03-14}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=McCollough|first1=Cynthia H.|last2=Bushberg|first2=Jerrold T.|last3=Fletcher|first3=Joel G.|last4=Eckel|first4=Laurence J.|date=2015-10-01|title=Answers to Common Questions About the Use and Safety of CT Scans|url=https://www.mayoclinicproceedings.org/article/S0025-6196(15)00591-1/abstract|journal=Mayo Clinic Proceedings|language=English|volume=90|issue=10|pages=1380–1392|doi=10.1016/j.mayocp.2015.07.011|issn=0025-6196|pmid=26434964|doi-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.medicalnewstoday.com/articles/306067.php|title=No evidence that CT scans, X-rays cause cancer|date=4 February 2016|website=Medical News Today|language=en|access-date=2019-03-14}}&lt;/ref&gt;

One study estimated that as many as 0.4% of cancers in the United States resulted from CT scans, and that this may have increased to as much as 1.5 to 2% based on the rate of CT use in 2007.&lt;ref name=&quot;Brenner2007&quot; /&gt; Others dispute this estimate,&lt;ref&gt;{{Cite journal|last1=Kalra|first1=Mannudeep K.|last2=Maher|first2=Michael M.|last3=Rizzo|first3=Stefania|last4=Kanarek|first4=David|last5=Shephard|first5=Jo-Anne O.|date=April 2004|title=Radiation exposure from Chest CT: Issues and Strategies|journal=Journal of Korean Medical Science|volume=19|issue=2|pages=159–166|doi=10.3346/jkms.2004.19.2.159|issn=1011-8934|pmc=2822293|pmid=15082885}}&lt;/ref&gt; as there is no consensus that the low levels of radiation used in CT scans cause damage. Lower radiation doses are used in many cases, such as in the investigation of renal colic.&lt;ref&gt;{{Cite journal|last1=Rob|first1=S.|last2=Bryant|first2=T.|last3=Wilson|first3=I.|last4=Somani|first4=B.K.|year=2017|title=Ultra-low-dose, low-dose, and standard-dose CT of the kidney, ureters, and bladder: is there a difference? Results from a systematic review of the literature|journal=Clinical Radiology|volume=72|issue=1|pages=11–15|doi=10.1016/j.crad.2016.10.005|pmid=27810168}}&lt;/ref&gt;

A person's age plays a significant role in the subsequent risk of cancer.&lt;ref name=&quot;Furlow2010&quot; /&gt; Estimated lifetime cancer mortality risks from an abdominal CT of a one-year-old is 0.1%, or 1:1000 scans.&lt;ref name=&quot;Furlow2010&quot; /&gt; The risk for someone who is 40 years old is half that of someone who is 20 years old with substantially less risk in the elderly.&lt;ref name=&quot;Furlow2010&quot; /&gt; The [[w:International Commission on Radiological Protection|International Commission on Radiological Protection]] estimates that the risk to a fetus being exposed to 10 [[w:mGy|mGy]] (a unit of radiation exposure) increases the rate of cancer before 20 years of age from 0.03% to 0.04% (for reference a CT pulmonary angiogram exposes a fetus to 4&amp;nbsp;mGy).&lt;ref name=&quot;Risk2011&quot; /&gt; A 2012 review did not find an association between medical radiation and cancer risk in children noting however the existence of limitations in the evidences over which the review is based.&lt;ref&gt;{{Cite journal|date=January 2012|title=[Diagnostic radiation exposure in children and cancer risk: current knowledge and perspectives]|journal=Archives de Pédiatrie|volume=19|issue=1|pages=64–73|doi=10.1016/j.arcped.2011.10.023|pmid=22130615|vauthors=Baysson H, Etard C, Brisse HJ, Bernier MO}}&lt;/ref&gt;

CT scans can be performed with different settings for lower exposure in children with most manufacturers of CT scans as of 2007 having this function built in.&lt;ref name=&quot;Semelka2007&quot; /&gt; Furthermore, certain conditions can require children to be exposed to multiple CT scans.&lt;ref name=&quot;Brenner2007&quot; /&gt; Current evidence suggests informing parents of the risks of pediatric CT scanning.&lt;ref name=&quot;pmid17646450&quot;&gt;{{Cite journal|date=August 2007|title=Informing parents about CT radiation exposure in children: it's OK to tell them|journal=Am J Roentgenol|volume=189|issue=2|pages=271–5|doi=10.2214/AJR.07.2248|pmid=17646450|vauthors=Larson DB, Rader SB, Forman HP, Fenton LZ|s2cid=25020619}}&lt;/ref&gt;

=== Risks vs benefits ===
The decision to request a CT scan involves an evaluation of potential benefits and associated risks. CT scans are invaluable for precise diagnoses, efficient treatment planning, and time-saving advantages across various medical contexts. While CT scans offer detailed anatomical insights for disease diagnosis and improved patient management, the use of ionizing radiation and contrast media raises concerns about potential harm and adverse effects. Every CT procedure must be justified, and the potential benefits should outweigh the associated risks. Preferential use of alternative, non-ionizing modalities such as ultrasound or MRI should be sought when diagnostically significant imaging can be obtained from them without compromising accuracy. In instances where a CT examination is deemed indispensable, diligent measures should be undertaken to optimize the procedure, ensuring the utilization of the minimum radiation dose essential for achieving diagnostically accurate scans. Adherence to the ALARA  guidelines becomes important to mitigate unnecessary radiation exposure, prioritizing the well-being of the patient.

== History ==
The history of X-ray computed tomography goes back to at least 1917 with the mathematical theory of the [[w:Radon transform|Radon transform]].&lt;ref name=&quot;Radon1917&quot;&gt;{{Cite book |last1=Thomas |first1=Adrian M. K. |url=https://books.google.com/books?id=zgezC3Osm8QC&amp;q=info:6gaBWGuVV0UJ:scholar.google.com/&amp;pg=PA5 |title=Classic Papers in Modern Diagnostic Radiology |last2=Banerjee |first2=Arpan K. |last3=Busch |first3=Uwe |date=2005-12-05 |publisher=Springer Science &amp; Business Media |isbn=978-3-540-26988-5 |language=en}}&lt;/ref&gt;&lt;ref name=&quot;pmid 18244009&quot;&gt;{{Cite journal |last=Radon J |date=1 December 1986 |title=On the determination of functions from their integral values along certain manifolds |journal=IEEE Transactions on Medical Imaging |volume=5 |issue=4 |pages=170–176 |doi=10.1109/TMI.1986.4307775 |pmid=18244009 |s2cid=26553287}}&lt;/ref&gt; In October 1963, [[w:William H. Oldendorf|William H. Oldendorf]] received a U.S. patent for a &quot;radiant energy apparatus for investigating selected areas of interior objects obscured by dense material&quot;.&lt;ref name=&quot;Oldendorf1978&quot;&gt;{{Cite journal |last=Oldendorf WH |year=1978 |title=The quest for an image of brain: a brief historical and technical review of brain imaging techniques |journal=Neurology |volume=28 |issue=6 |pages=517–33 |doi=10.1212/wnl.28.6.517 |pmid=306588 |s2cid=42007208}}&lt;/ref&gt; The first commercially viable CT scanner was invented by [[w:Godfrey Hounsfield|Godfrey Hounsfield]] in 1972.&lt;ref name=&quot;Richmond2004&quot;&gt;{{Cite journal |last=Richmond |first=Caroline |year=2004 |title=Obituary – Sir Godfrey Hounsfield |journal=BMJ |volume=329 |issue=7467 |pages=687 |doi=10.1136/bmj.329.7467.687 |pmc=517662}}&lt;/ref&gt; The 1979 [[w:Nobel Prize in Physiology or Medicine|Nobel Prize in Physiology or Medicine]] was awarded jointly to South African-American physicist [[w:Allan MacLeod Cormack|Allan MacLeod Cormack]] and British electrical engineer [[w:Godfrey Hounsfield|Godfrey Hounsfield]] &quot;for the development of computer-assisted tomography&quot;.&lt;ref&gt;{{Cite journal|last=Di Chiro|first=Giovanni|last2=Brooks|first2=Rodney A.|date=1979-11-30|title=The 1979 Nobel Prize in Physiology or Medicine|url=http://dx.doi.org/10.1126/science.386516|journal=Science|volume=206|issue=4422|pages=1060–1062|doi=10.1126/science.386516|issn=0036-8075}}&lt;/ref&gt;

=== Etymology ===
The word &quot;tomography&quot; is derived from the [[w:Ancient Greek|Greek]] ''tome'' (slice) and ''graphein'' (to write).&lt;ref&gt;{{Cite book |last=[[Frank Natterer]] |title=The Mathematics of Computerized Tomography (Classics in Applied Mathematics) |publisher=Society for Industrial and Applied Mathematics |year=2001 |isbn=978-0-89871-493-7 |pages=8}}&lt;/ref&gt; Computed tomography was originally known as the &quot;EMI scan&quot; as it was developed in the early 1970s at a research branch of [[w:EMI|EMI]], a company best known today for its music and recording business.&lt;ref&gt;{{Cite book |last=Sperry |first=Len |url=https://books.google.com/books?id=NzgVCwAAQBAJ&amp;q=Computed+tomography+was+originally+known+as+the+%22EMI+scan%22&amp;pg=PA259 |title=Mental Health and Mental Disorders: An Encyclopedia of Conditions, Treatments, and Well-Being [3 volumes]: An Encyclopedia of Conditions, Treatments, and Well-Being |date=2015-12-14 |publisher=ABC-CLIO |isbn=978-1-4408-0383-3 |page=259 |language=en}}&lt;/ref&gt; It was later known as ''computed axial tomography'' (''CAT'' or ''CT scan'') and ''body section röntgenography''.&lt;ref&gt;{{Cite journal |last=Hounsfield |first=G. N. |date=1977 |title=The E.M.I. Scanner |journal=Proceedings of the Royal Society of London. Series B, Biological Sciences |volume=195 |issue=1119 |pages=281–289 |bibcode=1977RSPSB.195..281H |doi=10.1098/rspb.1977.0008 |issn=0080-4649 |jstor=77187 |pmid=13396 |s2cid=34734270}}&lt;/ref&gt; The term &quot;CAT scan&quot; is no longer used because current CT scans enable for multiplanar reconstructions. This makes &quot;CT scan&quot; the most appropriate term, which is used by [[w:radiologist|radiologist]]s in common vernacular as well as in textbooks and scientific papers.&lt;ref&gt;{{Cite web |title=Difference Between CT Scan and CAT Scan {{!}} Difference Between |url=http://www.differencebetween.net/science/health/difference-between-ct-scan-and-cat-scan/ |access-date=2021-03-19 |language=en-US}}&lt;/ref&gt;&lt;ref&gt;{{Cite book |url=https://books.google.com/books?id=FqDUtcmUG-UC&amp;q=cat+scanner+term+was+used+earlier |title=Conquer Your Headaches |publisher=International Headache Management |year=1994 |isbn=978-0-9636292-5-8 |pages=115}}&lt;/ref&gt; In [[w:Medical Subject Headings|Medical Subject Headings]] (MeSH), &quot;computed axial tomography&quot; was used from 1977 to 1979, but the current indexing explicitly includes &quot;X-ray&quot; in the title.&lt;ref&gt;{{Cite web |title=MeSH Browser |url=https://meshb.nlm.nih.gov/record/ui?ui=D014057 |website=meshb.nlm.nih.gov}}&lt;/ref&gt;

== Society and culture ==
=== Campaigns ===
In response to increased concern by the public and the ongoing progress of best practices, the Alliance for Radiation Safety in Pediatric Imaging was formed within the [[w:Society for Pediatric Radiology|Society for Pediatric Radiology]]. In concert with the [[w:American Society of Radiologic Technologists|American Society of Radiologic Technologists]], the [[w:American College of Radiology|American College of Radiology]] and the [[w:American Association of Physicists in Medicine|American Association of Physicists in Medicine]], the Society for Pediatric Radiology developed and launched the Image Gently Campaign which is designed to maintain high-quality imaging studies while using the lowest doses and best radiation safety practices available on pediatric patients.&lt;ref&gt;{{Cite web |title=Image Gently |url=http://www.pedrad.org/associations/5364/ig/?page=365 |url-status=dead |archive-url=https://web.archive.org/web/20130609063515/http://www.pedrad.org/associations/5364/ig/?page=365 |archive-date=9 June 2013 |access-date=19 July 2013 |publisher=The Alliance for Radiation Safety in Pediatric Imaging}}&lt;/ref&gt; This initiative has been endorsed and applied by a growing list of various professional medical organizations around the world and has received support and assistance from companies that manufacture equipment used in Radiology.

Following upon the success of the ''Image Gently'' campaign, the American College of Radiology, the Radiological Society of North America, the American Association of Physicists in Medicine and the American Society of Radiologic Technologists have launched a similar campaign to address this issue in the adult population called ''Image Wisely''.&lt;ref&gt;{{Cite web |title=Image Wisely |url=http://www.imagewisely.org/ |url-status=dead |archive-url=https://web.archive.org/web/20130721032437/http://imagewisely.org/ |archive-date=21 July 2013 |access-date=19 July 2013 |publisher=Joint Task Force on Adult Radiation Protection}}&lt;/ref&gt;

The [[w:World Health Organization|World Health Organization]] and [[w:International Atomic Energy Agency|International Atomic Energy Agency]] (IAEA) of the United Nations have also been working in this area and have ongoing projects designed to broaden best practices and lower patient radiation dose.&lt;ref&gt;{{Cite web |title=Optimal levels of radiation for patients |url=http://new.paho.org/hq10/index.php?option=com_content&amp;task=view&amp;id=3365&amp;Itemid=2164 |url-status=dead |archive-url=https://web.archive.org/web/20130525051814/http://new.paho.org/hq10/index.php?option=com_content&amp;task=view&amp;id=3365&amp;Itemid=2164 |archive-date=25 May 2013 |access-date=19 July 2013 |publisher=World Health Organization}}&lt;/ref&gt;

=== Prevalence ===
&lt;div style=&quot;float:right; width:23em; height:20em; overflow:auto; border:0px&quot;&gt;
{|class=&quot;wikitable&quot;
|+{{nowrap|Number of  CT scanners by country (OECD)}}&lt;br /&gt;as of 2017&lt;ref&gt;{{Cite web |title=Computed tomography (CT) scanners |url=https://data.oecd.org/healtheqt/computed-tomography-ct-scanners.htm |publisher=OECD}}&lt;/ref&gt;&lt;br /&gt;(per million population)
!Country !! Value
|-
|{{flagcountry| JPN}} || 111.49
|-
|{{flagcountry| AUS}} || 64.35
|-
|{{flagcountry| ISL}} || 43.68
|-
|{{flagcountry| USA}} || 42.64
|-
|{{flagcountry| DNK}} || 39.72
|-
|{{flagcountry| CHE}} || 39.28
|-
|{{flagcountry| LVA}} || 39.13
|-
|{{flagcountry| KOR}} || 38.18
|-
|{{flagcountry| DEU}} || 35.13
|-
|{{flagcountry| ITA}} || 34.71
|-
|{{flagcountry| GRC}} || 34.22
|-
|{{flagcountry| AUT}} || 28.64
|-
|{{flagcountry| FIN}} || 24.51
|-
|{{flagcountry| CHL}} || 24.27
|-
|{{flagcountry| LTU}} || 23.33
|-
|{{flagcountry| IRL}} || 19.14
|-
|{{flagcountry| ESP}} || 18.59
|-
|{{flagcountry| EST}} || 18.22
|-
|{{flagcountry| FRA}} || 17.36
|-
|{{flagcountry| SVK}} || 17.28
|-
|{{flagcountry| POL}} || 16.88
|-
|{{flagcountry| LUX}} || 16.77
|-
|{{flagcountry| NZL}} || 16.69
|-
|{{flagcountry| CZE}} || 15.76
|-
|{{flagcountry| CAN}} || 15.28
|-
|{{flagcountry| SVN}} || 15.00
|-
|{{flagcountry| TUR}} || 14.77
|-
|{{flagcountry| NLD}} || 13.48
|-
|{{flagcountry| RUS}} || 13.00
|-
|{{flagcountry| ISR}} || 9.53
|-
|{{flagcountry| HUN}} || 9.19
|-
|{{flagcountry| MEX}} || 5.83
|-
|{{flagcountry| COL}} || 1.24
|-
|}
&lt;/div&gt;
Use of CT has increased dramatically over the last two decades.&lt;ref name=&quot;Smith2009&quot; /&gt; An estimated 72&amp;nbsp;million scans were performed in the United States in 2007,&lt;ref name=&quot;Berrington2009&quot; /&gt; accounting for close to half of the total per-capita dose rate from radiologic and nuclear medicine procedures.&lt;ref&gt;{{Cite journal |last1=Fred A. Mettler Jr |last2=Mythreyi Bhargavan |last3=Keith Faulkner |last4=Debbie B. Gilley |last5=Joel E. Gray |last6=Geoffrey S. Ibbott |last7=Jill A. Lipoti |last8=Mahadevappa Mahesh |last9=John L. McCrohan |last10=Michael G. Stabin |last11=Bruce R. Thomadsen |last12=Terry T. Yoshizumi |year=2009 |title=Radiologic and Nuclear Medicine Studies in the United States and Worldwide: Frequency, Radiation Dose, and Comparison with Other Radiation Sources — 1950-2007 |journal=Radiology |volume=253 |pages=520–531 |doi=10.1148/radiol.2532082010 |pmid=19789227 |number=2}}&lt;/ref&gt; Of the CT scans, six to eleven percent are done in children,&lt;ref name=&quot;Risk2011&quot; /&gt; an increase of seven to eightfold from 1980.&lt;ref name=&quot;Furlow2010&quot; /&gt; Similar increases have been seen in Europe and Asia.&lt;ref name=&quot;Furlow2010&quot; /&gt;  In Calgary, Canada, 12.1% of people who present to the emergency with an urgent complaint received a CT scan, most commonly either of the head or of the abdomen. The percentage who received CT, however, varied markedly by the [[w:emergency physician|emergency physician]] who saw them from 1.8% to 25%.&lt;ref&gt;{{Cite journal |last=Andrew Skelly |date=Aug 3, 2010 |title=CT ordering all over the map |journal=The Medical Post}}&lt;/ref&gt; In the emergency department in the United States, CT or [[w:Magnetic resonance imaging|MRI]] imaging is done in 15% of people who present with [[w:injuries|injuries]] as of 2007 (up from 6% in 1998).&lt;ref&gt;{{Cite journal |vauthors=Korley FK, Pham JC, Kirsch TD |date=October 2010 |title=Use of advanced radiology during visits to US emergency departments for injury-related conditions, 1998–2007 |journal=JAMA |volume=304 |issue=13 |pages=1465–71 |doi=10.1001/jama.2010.1408 |pmid=20924012 |doi-access=free}}&lt;/ref&gt;

The increased use of CT scans has been the greatest in two fields: screening of adults (screening CT of the lung in smokers, virtual colonoscopy, CT cardiac screening, and whole-body CT in asymptomatic patients) and CT imaging of children. Shortening of the scanning time to around 1 second, eliminating the strict need for the subject to remain still or be sedated, is one of the main reasons for the large increase in the pediatric population (especially for the diagnosis of [[w:appendicitis|appendicitis]]).&lt;ref name=&quot;Brenner2007&quot; /&gt; As of 2007, in the United States a proportion of CT scans are performed unnecessarily.&lt;ref name=&quot;Semelka2007&quot;&gt;{{Cite journal |vauthors=Semelka RC, Armao DM, Elias J, Huda W |date=May 2007 |title=Imaging strategies to reduce the risk of radiation in CT studies, including selective substitution with MRI |journal=J Magn Reson Imaging |volume=25 |issue=5 |pages=900–9 |doi=10.1002/jmri.20895 |pmid=17457809 |s2cid=5788891}}&lt;/ref&gt; Some estimates place this number at 30%.&lt;ref name=&quot;Risk2011&quot; /&gt; There are a number of reasons for this including: legal concerns, financial incentives, and desire by the public.&lt;ref name=&quot;Semelka2007&quot; /&gt; For example, some healthy people avidly pay to receive full-body CT scans as [[w:screening (medicine)|screening]]. In that case, it is not at all clear that the benefits outweigh the risks and costs. Deciding whether and how to treat [[w:incidentaloma|incidentaloma]]s is complex, radiation exposure is not negligible, and the money for the scans involves [[w:opportunity cost|opportunity cost]].&lt;ref name=&quot;Semelka2007&quot; /&gt;

==Additional Information==

=== Acknowledgements ===
I would like to express my gratitude to Prof. Lalit Kumar Gupta (Rayat Bahra University) for their guidance and academic encouragement. 

=== Competing interests ===
The authors declare that they have no competing interests.

=== Ethics statement ===
No ethics statement necessary.

== References ==
{{reflist|refs=
{{Citation |title=Advanced Documentation Methods in Studying Corinthian Black-figure Vase Painting |date=2019 |url=https://www.chnt.at/wp-content/uploads/eBook_CHNT23_Karl.pdf |work=Proceedings of the 23rd International Conference on Cultural Heritage and New Technologies (CHNT23) |publication-place=Vienna, Austria |isbn=978-3-200-06576-5 |access-date=2020-01-09 |last2=Bayer |first2=Paul |author3-link=Hubert Mara |last3=Mara |first3=Hubert |last4=Márton |first4=András |given1=Stephan |surname1=Karl}}&lt;/ref&gt;}}</text>
      <sha1>942ebxh72xn10f15wgw4nbve9hljuc5</sha1>
    </revision>
  </page>
  <page>
    <title>Information is a public good: Designing experiments to improve government</title>
    <ns>0</ns>
    <id>296910</id>
    <revision>
      <id>2622965</id>
      <parentid>2622388</parentid>
      <timestamp>2024-04-26T05:29:31Z</timestamp>
      <contributor>
        <username>DavidMCEddy</username>
        <id>218607</id>
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      <comment>/* Public goods */ wdsmth</comment>
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      <text bytes="140500" xml:space="preserve">{{Research project}} 

:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].''  

::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' 

== Abstract == 

This article reviews literature relevant to the claim that &quot;information is a public good&quot; and recommends experiments to quantify the impact of news on society, including on violent conflicts and broadly shared economic growth. We propose randomized controlled trials to evaluate the relative effectiveness of alternative interventions on the lethality of conflict and broadly shared economic growth. Experimental units would be polities in conflict or with incomes (nominal Gross Domestic Products, GDPs or gross local products) small enough so competitive local news outlets could be funded by philanthropies or organizations like the World Bank but large enough that their political economies have been tracked with sufficient accuracy to allow them to be considered in such an experiment. One factor in such experiments would be subsidies for local journalism, perhaps distributed to local news outlets on the basis of local elections, similar to the proposal of McChesney and Nichols (2021, 2022).

== Introduction ==

:''Information is a public good.''&lt;ref&gt;This is the title of Cagé and Huet (2021, in French).  However, the thrust of their book is very different.  It is subtitled, &quot;Refounding media ownership&quot;.  Their focus is on creating legal structure(s) to support journalistic independence as outlined in Cagé (2016).&lt;/ref&gt; 
:''Misinformation is a public nuisance.''&lt;ref&gt;The Wikipedia article on &quot;[[w:Public nuisance|Public nuisance]]&quot; says, &quot;In English criminal law, public nuisance was a common law offence in which the injury, loss, or damage is suffered by the public, in general, rather than an individual, in particular.&quot; (accessed 2023-04-24.)&lt;/ref&gt;
:''Disinformation is a public evil.''&lt;ref&gt;The initial author of this essay is unaware of any previous use of the term, &quot;public evil&quot;, but it seems appropriate in this context to describe content disseminated by mass media, including social media, curated with the explicit intent to convince people to support public policies contrary to the best interests of the audience and the general public.&lt;/ref&gt; 

=== Public goods ===

In economics, a [[w:public good|public good]] is a good (or service) that is both [[w:non-rivalrous|non-rivalrous]] and [[w:non-excludable|non-excludable]].&lt;ref&gt;e.g., Cornes and Sandler (1996).&lt;/ref&gt; Non-rivalrous means that we can all consume it at the same time. An apple is rivalrous, because if I eat an apple, you cannot eat the same apple. 

A printed newspaper may be rivalrous, because it may not be easy for you and me to hold the same sheet of paper and read it at the same time. However, the ''news'' itself is non-rivalrous, because both of us and anyone else can consume the same news at the same time, once it is produced, especially if it's published openly on the Internet or broadcasted on radio or television. 

Non-excludable means that once the good is produced, anyone can use it without paying for it. Information is non-excludable, because everyone can consume it at the same time once it becomes available. [[w:Copyright|Copyright]] law does ''not'' apply to information: It applies to ''expression''.&lt;ref&gt;The US Copyright Act of 1976, Section 102, says, &quot;Copyright protection subsists ... in original works of authorship fixed in any tangible medium of expression ... . In no case does copyright protection ... extend to any idea, procedure, process, system, method of operation, concept, principle, or discovery.&quot; 17 U.S. Code § 102. &lt;!--US Copyright Law of 1976--&gt;{{cite Q|Q3196755}}&lt;/ref&gt; [[w:Joseph Stiglitz|Stiglitz]] (1999) said that [[w:Thomas Jefferson|Thomas Jefferson]] anticipated the modern concept of information as a public good by saying, &quot;He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me.&quot; Stiglitz distinguished between &quot;push and pull mechanisms&quot; to promote innovation and creative work: &quot;Push&quot; mechanisms pay for work upfront, hoping that it will achieve a desired outcome, like citizen-directed subsidies for newspapers. &quot;Pull&quot; mechanisms set a target and then reward those who reach the target, like copyrights and patents.&lt;ref&gt;Baker (2023).&lt;/ref&gt;

Lindahl (1919, 1958) recommended taxing people for public goods in proportion to the benefits they receive. For subsidies for news, especially citizen-directed, this would mean taxing primarily the poor and middle class to fund this.&lt;ref&gt;For more on this, see the Wikipedia articles on [[w:Lindahl tax|Lindahl tax]] and [[w:Theories of taxation|Theories of taxation]].&lt;/ref&gt; If they receive benefits as claimed in the literature cited in this article, the benefits they receive would soon exceed the taxes they pay for it, making the news subsidies effectively free in perpetuity, paid by benefits the poor and middle class would not have without these subsidies. If Piketty (2021, cited below with Figure 1) is correct, the ultra-wealthy would likely also benefit in absolute terms, though the relative distinction between them and the poor would be reduced.

This article recommends [[w:randomized controlled trials|randomized controlled trials]] to quantify the extent to which experimental interventions benefit the public by modifying information environment(s) in ways that (a) reduce political polarization and / or violence and / or (b) improve broadly shared peace and prosperity for the long term.

=== Sharing increases the value === 

The logic behind claiming that &quot;information is a public good&quot; can be easily understood as follows:  

:''If I know the best solution to any major societal problem, it will not help anyone unless a critical mass of some body politic shares that perception.  Conversely, if a critical mass of a body politic believes in the need to implement a certain reform, it will happen, even if I am ignorant of it or completely opposed to it.''  

We can extend this analysis to our worst enemies:  

:It is in ''our best interest'' to help people supporting our worst enemies get information they want, ''independent'' of controls that people with power exercise over nearly all major media today:  If our actions reduce the ability of their leaders to censor their media (and of our political and economic leaders to censor ours), the information everyone gets should make it harder for leaders to convince others to support measures contrary to nearly everyone's best interests.  

Everyone will benefit, except perhaps the people who currently control most of the money for the media.&lt;ref&gt;The power relationship between media and politicians can go both ways.  In addition to asking the extent to which politicians control the media, we can also consider the extent to which political leaders might feel constrained by the major media: To what extent do the major media create the stage upon which politicians read their lines, as claimed in the Wikiversity article on &quot;[[Confirmation bias and conflict]]&quot;?  Might a more diverse media environment make it easier for political leaders to pursue policies informed more by available research and less by propaganda?  Might experiments as described herein help politicians develop more effective governmental policies, because of a reduction in the power of media whose ownership and funding are more diverse? This is discussed further in this article in a section on [[Information is a public good: Designing experiments to improve government #Media and war|Media and war]].&lt;/ref&gt;

=== World Bank on the value of information === 

In 2002 the President of the [[w:World Bank|World Bank]],&lt;ref&gt;The 2022 World Bank Group portfolio was 104 billion USD (World Bank 2022, Table 1, p. 13; 17/116 in PDF).  An improvement of 0.1 percentage points in the performance of that portfolio would be 104 million.  A lot could be accomplished with budgets much smaller than this.&lt;/ref&gt; [[w:James Wolfensohn|James Wolfensohn]], wrote, &quot;[A] free press is not a luxury. It is at the core of equitable development. The media can expose corruption. ... They can facilitate trade [and bring] health and education information to remote villages ... . But ... the independence of the media can be fragile and easily compromised. All too often governments shackle the media. Sometimes control by powerful private interests restricts reporting. ... [T]o support development, media need the right environment{{mdash}}in terms of freedoms, capacities, and checks and balances.&quot;&lt;ref&gt;Wolfensohn (2002). More on this is available in other contributions to Islam et al. (2002) including [[w:Joseph Stiglitz|Stiglitz]] (2002), who noted the following: &quot;There is a natural asymmetry of information between those who govern and those whom they are supposed to serve. ... Free speech and a free press not only make abuses of governmental powers less likely, they also enhance the likelihood that people's basic social needs will be met. ... [S]ecrecy distorts the arena of politics. ... Neither theory nor evidence provides much support for the hypothesis that fuller and timelier disclosure and discussion would have adverse effects. ... The most important check against abuses is a competitive press that reflects a variety of interests. ... [F]or government officials to appropriate the information that they have access to for private gain ... is as much theft as stealing any other public property.&quot;&lt;/ref&gt;

=== US Postal Service Act of 1792: a natural experiment === 

[[w:Robert W. McChesney|McChesney]] and [[w:John Nichols (journalist)|Nichols]] (2010, 2016) suggested that the US [[w:Postal Service Act|Postal Service Act]]&lt;ref&gt;Wikipedia &quot;[[w:Postal Service Act|Postal Service Act]]&quot;, accessed 2023-07-11.&lt;/ref&gt; of 1792 made a major contribution to making the US what it is today.  Under that act, newspapers were delivered up to 100 miles for a penny, when first class postage was between 6 and 25 cents depending on distance.  They estimated that between 1840 and 1844, the US postal subsidy was 0.211 percent of GDP with federal printing subsidies adding another 0.005 percent, totaling 0.216 percent of GDP,&lt;ref name=McC-N2010&gt;McChesney and Nichols (2010, pp. 310-311, note 88).&lt;/ref&gt; roughly $140 per person per year in 2019 dollars.&lt;ref name=McN_IMF&gt;International Monetary Fund (2023):  US Gross domestic product per capita at current prices was estimated at $65,077 for 2019 on 2023-04-28. 0.211% of $65,077 = $137; round to $140 for convenience.&lt;/ref&gt; We use 2019 dollars here to make it easy to compare with Rolnik et al. (2019), who recommended $50 per adult per year, which is roughly 0.08 percent of US GDP.  Rolnik et al. added that the level of subsidies would require &quot;extensive deliberation and experimentation&quot;.&lt;ref name=Rolnik&gt;Rolnik et al. (2019, p. 44).  Per [[w:Demographics of the United States]], 24 percent of the US population is under 18, so adults are 76 percent of the population. Thus, $50 per adult is $37.50 per capita. US GDP per capita in $65,0077 in 2019 in current dollars per International Monetary Fund (2023). Thus, $37.50 per capita would be roughly 0.077 percent of GDP; round to 0.08 percent for convenience.&lt;/ref&gt; More recently McChesney and Nichols have recommended 0.15 percent of GDP, considering the fact that the advent of the Internet has nearly eliminated the costs of printing and distribution.&lt;ref name=McC-N2021&gt;McChesney and Nichols (2021; 2022, p. 19).&lt;/ref&gt;

[[w:Alexis de Tocqueville|Tocqueville]], who visited the US in 1831, observed the following:  
* [T]he liberty of the press does not affect political opinion alone, but extends to all the opinions of men, and modifies customs as well as laws. ... I approve of it from a consideration more of the evils it prevents, than of the advantages it insures.&lt;ref&gt;Tocqueville (1835; 2001, p. 91).  In 2002 Roumeen Islam stated this more forcefully:  &quot;Arbitrary actions by government are always to be feared. If there is to be a bias in the quantity of information that is released, then erring on the side of more freedom rather than less would appear to cause less harm.&quot; (World Bank, 2002, pp. 21-22; 33-34/336 in pdf).&lt;/ref&gt;
* The liberty of writing ... is most formidable when it is a novelty; for a people who have never been accustomed to hear state affairs discussed before them, place implicit confidence in the first tribune who presents himself.  The Anglo-Americans have enjoyed this liberty ever since the foundation of the Colonies ... .  A glance at a French and an American newspaper is sufficient to show the difference ... .  In France, the space allotted to commercial advertisements is very limited, and the news-intelligence is not considerable; but the essential part of the journal is the discussion of the politics of the day.  In America, three-quarters of the enormous sheet are filled with advertisements, and the remainder is frequently occupied by political intelligence or trivial anecdotes:  it is only from time to time that one finds a corner devoted to passionate discussions, like those which the journalists of France every day give to their readers.&lt;ref&gt;Tocqueville (1835; 2001, p. 92).&lt;/ref&gt; 
* It has been demonstrated by observation, and discovered by the sure instinct even of the pettiest despots, that the influence of a power is increased in proportion as its direction is centralized.&lt;ref&gt;Tocqueville (1835; 2001, pp. 92-93).&lt;/ref&gt;
* [T]he number of periodical and semi-periodical publications in the United States is almost incredibly large. In America there is scarcely a hamlet which does not have its newspaper.&lt;ref&gt;Tocqueville (1835; 2001, p. 93).&lt;/ref&gt;
* In the United States, each separate journal exercises but little authority; but the power of the periodical press is second only to that of the people ... .&lt;ref&gt;Tocqueville (1835; 2001, p. 94). &lt;/ref&gt;

[[File:Real US GDP per capita in 5 epocs.svg|thumb|Figure 1.  Average annual income (Gross Domestic Product per capita adjusted for inflation) in the US 1790-2021 showing five epochs identified in a &quot;breakpoint&quot; analysis (to 1929, 1933, 1945, 1947, 2021) documented in the Wikiversity article on &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;.&lt;ref&gt;Wikiversity &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;, accessed 2023-07-18.&lt;/ref&gt; Piketty (2021, p. 139) noted, &quot;In the United States, the national income per inhabitant rose at a rate ... of 2.2 percent between 1950 to 1990 when the top tax rate reached on average 72 percent.  The top rate was then cut in half, with the announced objective of boosting growth.  But in fact, growth fell by half, reaching 1.1 percent per annum between 1990 and 2020&quot;.&lt;ref&gt;A more recent review of the literature of the impact of inequality on growth is provided by Jahangir (2023, sec. 3), who notes that some studies have claimed that inequality ''increases'' the rate of economic growth, while other reach the opposite conclusion. However, 'the preponderant academic position is shifting from the argument that “we don’t have enough evidence” and towards seriously addressing and combating economic inequality.'&lt;/ref&gt; Our analysis of US GDP per capita from Measuring Worth do not match Piketty's report exactly, but they are close. We got 2.3 percent annual growth from 1947 to 1990 then 1.8 percent to 2008 and 1.1 percent to 2020. However, we have so far been unable to find a model that suggests that this decline is statistically significant.]]  

To what extent was [[w:Alexis de Tocqueville|Tocqueville's]] &quot;incredibly large&quot; &quot;number of periodical and semi-periodical publications in the United States&quot; due to the US Postal Service Act of 1792? To what extent did that &quot;incredibly large&quot; number of publications encourage literacy, limit political corruption, and help the US of that day remain together and grow both in land area and economically while contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically? To what extent does the enormous power of the US today rest on the economic growth of that period and its impact on the political culture of that day continuing to the present?&lt;ref&gt;Wikiversity &quot;[[The Great American Paradox]]&quot;, accessed 2023-06-12.&lt;/ref&gt; That growth transformed the US into the world leader that it is today;  see Figure 1.  In the process, it generated new technologies that benefit the vast majority of the world's population alive today. If the newspapers Tocqueville read made any substantive contribution to the growth summarized in Figure 1, the information in those newspapers were public goods potentially benefiting the vast majority of humanity ''to the end of human civilization.''&lt;ref&gt;Acemoglu (2023) documents how the power of monopolies and other politically favored groups often distorts the direction of technology development into suboptimal technologies.  Might increasing the funding for more independent news outlets reduce the power of such favored groups and thereby help correct these distortions and deliver &quot;sizable welfare benefits&quot;, e.g., &quot;in the context of industrial automation, health care, and energy&quot;?&lt;/ref&gt;

=== Other economists === 

We cannot prove that the diversity of newspapers in the early US contributed to the economic growth it experienced. Banerjee and Duflo (2019) concluded that no one knows how to create economic growth. They won the 2019 Nobel Memorial Prize in Economics with Michael Kremer for their leadership in using [[w:randomized controlled trials|randomized controlled trials]]&lt;ref&gt;Wikipedia &quot;[[w:Randomized controlled trials|Randomized controlled trials]]&quot;, accessed 2023-07-11.&lt;/ref&gt; to learn how to reduce global poverty.&lt;ref&gt;Wikipedia &quot;[[w:2019 Nobel Memorial Prize in Economic Sciences|2019 Nobel Memorial Prize in Economic Sciences]]&quot;, accessed 2023-06-13.  Nobel Prize (2019).  Amazon.com indicates that distribution of the book started 2019-11-12, twenty-nine days after the Nobel prize announcement 2019-10-14. Evidently the book must have been completed before the announcement.&lt;/ref&gt; More recently, Wake et al. (2021) found evidence that ''the economic costs of curbing press freedom persist long after such freedoms have been restored.''&lt;ref&gt;See also Nguyen et al. (2021).&lt;/ref&gt; And Mohammadi et al. (2022) found that economic growth rates were impacted by civil liberties, economic and press freedom and the economic growth rates of neighbors (spacial autocorrelation) but not democracy. These findings of Mohammidi et al. (2022) and Wake et al. (2021) reinforce Thomas Jefferson's 1787 comment that, &quot;were it left to me to decide whether we should have a government without newspapers, or newspapers without a government, I should not hesitate a moment to prefer the latter.&quot;&lt;ref&gt;From a letter to Colonel Edward Carrington (16 January 1787), cited in Wikiquote, &quot;[[Wikiquote:Thomas Jefferson|Thomas Jefferson]]&quot;, accessed 2023-07-29.&lt;/ref&gt; To what extent might experiments like those recommended in this article either reinforce or refute this claim of Jefferson from 1787?

=== Randomized controlled trials to quantify the value of information === 

This article suggests randomized controlled trials to quantify the impact of citizen-directed subsidies for journalism, roughly following the recommendations of McChesney and Nichols (2021, 2022) to distribute some small percentage of GDP to local news nonprofits ''via local elections''.  Philanthropies could fund such experiments for some of the smallest and poorest places in the world.  Organizations like the World Bank could fund such experiments as adjuncts to a random selection from some list of other interventions they fund, justified for the same reason that they would not consider funding anything without appropriate accounting and auditing of expenditures, as discussed further below.  

Before making suggestions regarding experiments, we review previous research documenting how information might be a public good.

== Previous research ==

Before considering optimal level of subsidies for news, it may be useful to consider the research for which [[w: Daniel Kahneman|Daniel Kahneman won the 2002 Nobel Memorial Prize in Economics]].&lt;ref&gt;Wikipedia &quot;[[w:Daniel Kahneman|Daniel Kahneman]]&quot;, accessed 2023-04-28.&lt;/ref&gt; Most important for present purposes may be that virtually everyone: 

# thinks they know more than they do ([[w:Overconfidence effect|Overconfidence]]),&lt;ref&gt;Wikipedia &quot;[[w:Overconfidence effect|Overconfidence effect]]&quot;, accessed 2023-04-29. Kahneman and co-workers have documented that experts are also subject to overconfidence and in some cases may be worse.  Kahneman and Klein (2009) found that expert intuition can be learned from frequent, rapid, high-quality feedback about the quality of their judgments.  Unfortunately, few fields have that much quality feedback.  Kaheman et al. (2021) call practitioners with credentials but without such expert intuition &quot;respect-experts&quot;.  Kahneman (2011, p. 234) said his &quot;most satisfying and productive adversarial collaboration was with Gary Klein&quot;.&lt;/ref&gt; and 
# prefers information and sources consistent with preconceptions. ([[w:Confirmation bias|Confirmation bias]]).&lt;ref&gt;Wikipedia &quot;[[w:Confirmation bias|Confirmation bias]]&quot;, accessed 2023-04-29.&lt;/ref&gt;

To what extent do media organizations everywhere exploit the confirmation bias and overconfidence of their audience to please those who control the money for the media, and to what extent might this ''reduce'' broadly shared economic growth?  The proposed experiments should include efforts to quantify this, measuring, e.g., local incomes, inequality, political polarization and the impact of interventions attempting to improve such.  

Plous wrote, &quot;No problem in judgment and decision making is more prevalent and more potentially catastrophic than overconfidence.&quot;&lt;ref&gt;Plous (1993, p. 217).  See also Wikipedia &quot;[[w:Overconfidence effect|Overconfidence effect]]&quot;, accessed 2023-04-29.&lt;/ref&gt; It contributes to inordinate losses by all parties in negotiations of all kinds&lt;ref&gt;Thompson (2020).&lt;/ref&gt; including lawsuits,&lt;ref&gt;Loftus and Wagenaar (1988).&lt;/ref&gt; strikes,&lt;ref&gt;Babcock and Olson (1992) and Thompson and Loevenstein (1992).&lt;/ref&gt; financial market bubbles and crashes,&lt;ref&gt;Daniel et al. (1998).&lt;/ref&gt; and politics and international relations,&lt;ref&gt;&lt;!-- Dominic D.P. Johnson (2020) Strategic instincts: the adaptive advantages of cognitive biases in international politics--&gt;{{cite Q|Q120967807}}&lt;/ref&gt; including wars.&lt;ref&gt;Johnson (2004).&lt;/ref&gt;

Might the frequency and expense of lawsuits, strikes, financial market volatility, political coruption and wars be reduced by encouraging people to get more curious and search more often for information that might contradict their preconceptions?  Might such discussions be encouraged by interventions such as increasing the total funding for news through many small, independent, local news organizations?  If yes, to what extent might such experimental interventions threaten the hegemony of major media everywhere while benefitting everyone, with the possible exception of those who benefit from current systems of political corruption?  

[[File:Knowledge v. public media.png|thumb|Figure 2.  Knowledge v. public media: Percent correct answers in surveys of knowledge of domestic and international politics vs. per capita subsidies for public media in Denmark (DK), Finland (FI), the United Kingdom (UK) and the United States (US).&lt;ref&gt;&quot;politicalKnowledge&quot; dataset in Croissant and Graves (2022), originally from ch. 1, chart 8, p. 268 and ch. 4, chart 1, p. 274, McChesney and Nichols (2010).&lt;/ref&gt;]]

One attempt to quantify this appears in Figure 2, which summarizes a natural experiment on the impact of government subsidies for public media on public knowledge of domestic and international politics: Around 2008 the governments of the US, UK, Denmark and Finland provided subsidies of $1.35, $80, $101 and $101 per person per year, respectively, for public media.  A survey of public knowledge of domestic and international politics found that people with college degrees seemed to be comparably well informed in the different countries, but people with less education were better informed in the countries with higher public subsidies.  

Kaviani et al. (2022) studied the impact of &quot;the staggered expansion of [[w:Sinclair Broadcast Group|Sinclair Broadcast Group]]: the largest conservative network in the U.S.&quot; They documented a decline in Corporate Social Responsibility (CSR) ratings of firms headquartered in Sinclair expansion areas.  They also documented a &quot;right-ward ideological shift&quot; in coverage that was &quot;nearly one standard deviation of the ideology distribution&quot; as well as &quot;substantial decreases in coverage of local politics substituted by increases in national politics.&quot; Ellison (2024) said that &quot;Sinclair's recipe for TV news&quot; includes an annual survey asking viewers, &quot;What are you most afraid of?&quot; Sinclair reportedly focuses on that while implying in their coverage &quot;that America's cities, especially those run by Democratic politicians, are dangerous and dysfunctional.&quot; 

Sources in France are concerned that billionaire [[w:Vincent Bolloré|Vincent Bolloré]] has purchased a substantial portion of French media and used it effectively to promote the French far right.&lt;ref&gt;Francois (2022). Cagé (2022). Cagé and Stetler (2022).&lt;/ref&gt; Scheidler (2024a) reported that the concentration of ownership the German media &quot;has not yet reached the extreme forms observed in France, the United Kingdom or the United States, but the process of consolidation initiated several decades ago has transformed a landscape renowned for its decentralization.&quot;&lt;ref&gt;Translated from, &quot;la concentration de la propriété dans la presse suprarégionale n’a pas encore atteint les formes extrêmes observées en France, au Royaume-Uni ou aux États-Unis, mais le processus de consolidation enclenché depuis plusieurs décennies a transformé un paysage réputé pour sa décentralisation.&quot;&lt;/ref&gt;&lt;ref&gt;See also ''Die Tageszeitung'' (2023).&lt;/ref&gt; Scheidler (2024b) reported that there still exists a wide range of constructive media criticism in Germany, but it gets less coverage than before in the increasingly consolidated major media. This has driven many who are not happy with these changes to alternative media such as ''[[w:Die Tageszeitung|Die Tageszeitung]]'', founded in 1978.

Benton wrote that past research has shown that strong local newspapers &quot;increase voter turnout, reduce government corruption, make cities financially healthier, make citizens more knowledgeable about politics and more likely to engage with local government, force local TV to raise its game, encourage split-ticket (and thus less uniformly partisan) voting, make elected officials more responsive and efficient ... And ... you get to reap the benefits of all those positive outcomes ''even if you don’t read them yourself''.&quot;&lt;ref&gt;Benton (2019); italics in the original.  See also Green et al. (2023, p. 7), Schulhofer-Wohl and Garrido (2009) and Stearns and Schmidt (2022).  A not quite silly example of this is documented in the Wikipedia article on the &quot;[[w:City of Bell scandal|City of Bell scandal]]&quot; accessed 2023-05-05:  Around 1999 the local newspaper died.  In 2010 the ''[[w:Los Angeles Times|Los Angeles Times]]'' reported that the city was close to bankruptcy in spite of having atypically high property tax rates.  The compensation for the City Manager was almost four times that of the President of the US, even though Bell, California, had a population of only approximately 38,000.  The Chief of Police and most members of the City Council also had exceptionally high compensations.  It was as if the City Manager had said in 1999, &quot;Wow:  The watchdog is dead.  Let's have a party.&quot;&lt;/ref&gt;

We feel a need to repeat that last comment:  ''You and I'' benefit from others consuming news that we do not, because they become less likely to be stampeded into voting contrary to their best interests{{mdash}}and ours{{mdash}}and more likely to lobby effectively against questionable favors to major political campaign contributors or other people with power, underreported by major media that have conflicts of interest in balanced coverage of anything that might offend people with substantive control of their funding.  That suggests that everyone might benefit from subsidizing ''a broad variety of independent'' local news outlets consumed by others.&lt;ref&gt;Some of those who benefit from the current system of political corruption may lose from the increased transparency produced by increases in the quality, quantity, diversity, and broader consumption of news.  However, Bezruchka (2023) documents how even the ultra-wealthy in countries with high inequality generally have shorter life expectancies than their counterparts in more egalitarian societies:  What they might lose in social status would likely be balanced by a reduction in stress and exposure to life-threatening incidents.&lt;/ref&gt; Experiments along the lines discussed below could attempt to evaluate these claims and estimate their magnitudes.

== How fair is the US tax system? ==

How fair is the US tax system?  It depends on who is asked and how fairness is defined.  

[[File:Share of taxes vs. AGI.svg|thumb|Figure 3.  Effective tax rate vs. Adjusted Gross Income (AGI).&lt;ref&gt;York (2023) based on analyses published by the US Internal Revenue Service (IRS).&lt;/ref&gt;]]

The [[w:Tax Foundation|Tax Foundation]] computed the effective tax rate in different portions of the distribution of Adjusted Gross Income (AGI), plotted in Figure 3.  They noted that,&quot;half of taxpayers paid 99.7 percent of federal income taxes&quot;. The effective tax rate on the 1 percent highest adjusted gross incomes (AGIs) was 26 percent, almost double (1.91 times) the average, while the effective tax rate for the bottom half was 3.1 percent, only 23 percent of the average.&lt;ref&gt;York (2023).&lt;/ref&gt;

The Tax Foundation did ''not'' mention that we get a very different perspective from considering ''gross income'' rather than AGI. Leiserson and Yagan (2021)&lt;ref&gt;published by the Biden White House.&lt;/ref&gt; estimated that the average ''effective'' federal individual income tax rate paid by America’s 400 wealthiest families&lt;ref&gt;The &quot;400 wealthiest families&quot; are identified in &quot;[[w:The Forbes 400|The Forbes 400]]&quot;; see the Wikipedia article with that title, accessed 2023-05-07.&lt;/ref&gt; was between 6 and 12 percent with the most likely number being 8.2 percent. The difference comes in the ''adjustments'', while the uncertainty comes primarily from appreciation in the value of unsold stock,&lt;ref&gt;Unsold stock or other property subject to capital gains tax, which in 2022 was capped at 20 percent;  see Wikipedia, &quot;[[w:Capital gains tax in the United States|Capital gains tax in the United States]]&quot;, accessed 2023-05-08.&lt;/ref&gt; which is taxed at a maximum of 20 percent when sold and never taxed if passed as inheritance.&lt;ref&gt;The Wikipedia article on &quot;[[w:Estate tax in the United States|Estate tax in the United States]]&quot; describes an &quot;Exclusion amount&quot;, which is not taxed in inheritance.  That exclusion amount was $675,000 in 2001 and has generally trended upwards since except for 2010, and was $12.06 million in 2022. (accessed 2023-05-08.)&lt;/ref&gt;

Divergent claims about ''business'' taxes can similarly be found.  Watson (2022) claimed  that, &quot;Corporate taxes are one of the most economically damaging ways to raise revenue and are a promising area of reform for states to increase competitiveness and promote economic growth, benefiting both companies and workers.&quot;  His &quot;economically damaging&quot; claim seems contradicted by the claim of Piketty (2021, p. 139), cited with Figure 1 above, that when the top income tax rate was cut in half, the rate of economic growth in the US fell by half, instead of increasing as Watson (2022) suggested. By contrast, Fuhrmann and Uradu (2023) describe, &quot;How large corporations avoid paying taxes&quot;.   

[[File:UStaxWords.svg|thumb|Figure 4.  Millions of words in the US federal tax code and regulations, 1955-2015, according to the [[w:Tax Foundation|Tax Foundation]]. [1=income tax code; 2=other tax code; 3=income tax regulations; 4=other tax regulations; solid line= total]&lt;ref&gt;&quot;UStaxWords&quot; dataset in Croissant and Graves (2022) from the Tax Foundation.&lt;/ref&gt;]]

One reference on the difference between &quot;adjusted&quot; and &quot;gross&quot; income is US federal tax code and regulations, which grew from 1.4 million words in 1955 to over 10 million in 2015, averaging 145,000 additional words each year;  see Figure 4.  

How does this relate to media?

== How do media organizations make money? ==

Media organizations everywhere sell changes in audience behaviors to the people who give them money.  If they do not have an audience, they have nothing to sell.  If they sufficiently offend their funders, they will not get the revenue needed to produce content.&lt;ref&gt;A famous illustration of this conflict between content and funding was when CBS Chairman [[w:William S. Paley|William Paley]] reportedly told [[w:Edward R. Murrow|Edward R. Murrow]] in 1958 that he was discontinuing Murrow's award-winning show ''[[w:See It Now|See It Now]]'', because &quot;I don't want this constant stomach ache every time you do a controversial subject&quot;, documented in Friendly (1967, p. 92).&lt;/ref&gt;  

The major media in the US have conflicts of interest in honestly reporting on discussions in congress on copyright law or on anything that might impact a major advertiser or might make it easier for politicians to get elected by spending less money on advertising.  McChesney (2015) insisted that the major media are not interested in providing information that people want:  They are interested in making money and protecting the interests of the ultra-wealthy, who control the largest advertising budgets.  For example, media coverage of the roughly 40,000 people who came to [[w:1999 Seattle WTO protests|Seattle in 1999 to protest the WTO]] Ministerial Conference there&lt;ref&gt;Wikipedia &quot;[[w:1999 Seattle WTO protests|1999 Seattle WTO protests]]&quot;, accessed 2023-05-08.&lt;/ref&gt; and the 10,000 - 15,000 who came to [[w:Washington A16, 2000|Washington, DC, the following year]] to protest the International Monetary Fund and the World Bank,&lt;ref&gt;Wikipedia &quot;[[w:Washington A16, 2000|Washington A16, 2000]]&quot;, accessed 2023-05-08.&lt;/ref&gt; included &quot;some outstanding pieces produced by the corporate media, but those were exceptions to the rule. ... [T]he closer a story gets to corporate power and corporate domination of our society, the less reliable the corporate news media is.&quot;&lt;ref&gt;McChesney (2015, p. xx).&lt;/ref&gt; 

Aaron (2021) said, &quot;Bob McChesney ... taught me [to] look at ... the stories that are cheap to cover.&quot; Between around 1975 and 2000, the major commercial broadcasters in the US fired nearly all their investigative journalists&lt;ref&gt;McChesney (2004, p. 81):  &quot;A five-year study of investigative journalism on TV news completed in 2002 determined that investigative journalism has all but disappeared from the nation's commercial airwaves.&quot;&lt;/ref&gt; and replaced them with the police blotter.  It's easy and cheap to repeat what the police say.&lt;ref&gt;Holmes (2022) quoted Ryan Sorrell, Founder and Publisher of the Kansas City Defender, as saying, &quot;the media often parrots or repeats what police and news releases say.&quot;&lt;/ref&gt; A news outlet can do that without seriously risking loss of revenue.  In addition, poor defendants who may not have money for legal defense will rarely have money to sue a media outlet for defamation.  By contrast, a news report on questionable activities by a major funder risks both direct loss of advertising revenue and being sued.&lt;ref&gt;The risks of being sued include the risks of [[w:Strategic lawsuit against public participation|strategic lawsuits against public participation]] (SLAPPs) by major organizations, which can intimidate journalists and publishers as well as potential whistleblowers, who might inform journalists of violations of law by their employers.  Some of these are documented in the &quot;[[w:Freedom of the press in the United States#U.S. Press Freedom Tracker|U.S. Press Freedom Tracker]]&quot;, maintained by the [[w:Freedom of the Press Foundation|Freedom of the Press Foundation]] and the [[w:Committee to Protect Journalists|Committee to Protect Journalists]].  These include arrests, assaults, threats, denial of access, equipment damage, prior restraint, and subpoenas which could intimidate journalists, publishers, and employees feeling a need to expose violations of law and threats to public safety.  See Wikipedia &quot;[[w:Freedom of the Press Foundation|Freedom of the Press Foundation]]&quot;, &quot;[[w:Committee to Protect Journalists|Committee to Protect Journalists]]&quot;, and &quot;[[w:Strategic lawsuit against public participation|Strategic lawsuit against public participation]]&quot;, accessed 2023-07-11.&lt;/ref&gt;  These risks impose a higher standard of journalism (and additional costs) when reporting on questionable activities by people with power than when reporting on poor people.  This is a much bigger problem in countries where libel is a criminal rather than a civil offense or where truth is not a defense for libel.&lt;ref&gt;Islam et al. (2002), esp. pp. 12-13 (24-25/336 in pdf), p. 50 (62/336 in pdf), and ch. 11, pp. 207-224 (219-236/336 in pdf). [[w:United States defamation law|Truth was not a defense against libel in the US]] in 1804 when Harry Croswell lost in ''[[w:United States defamation law#People v. Croswell|People v. Croswell]]''.  That began to change the next year when the [[w:United States defamation law#People v. Croswell|New York State Legislature]] changed the law to allow truth as a defense against a libel charge. Seventy years earlier in 1735 [[w:John Peter Zenger#Libel case|John Peter Zenger]] was acquitted of a libel charge, but only by [[w:Jury nullification in the United States|jury nullification]].&lt;/ref&gt;

[[File:U.S. incarceration rate since 1925.svg|thumb|Figure 5.  Percent of the US population in state and federal prisons  [male (dashed red), combined (solid black), female (dotted green)]&lt;ref&gt;&quot;USincarcerations&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

After about 1975 the public noticed the increased coverage of crime in the broadcast news and concluded that crime was out of control, when there had been no substantive change in crime.  They voted in a generation of politicians, who promised to get tough on crime.  The incarceration rate in the US went from 0.1 percent to 0.5 percent in the span of roughly 25 years, after having been fairly stable for the previous 50 years;  see Figure 5.&lt;ref&gt;Potter and Kapeller (1998). Sacco (1998, 2005).&lt;/ref&gt;  

[[File:IncomeInequality9b.svg|thumb|Figure 6.  Average and quantiles of family income (Gross Domestic Product per family) in constant 2010 dollars.&lt;ref&gt;&quot;incomeInequality&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

Around that same time, income inequality in the US began to rise;  see Figure 6.&lt;ref&gt;Bezruchka (2023) summarizes research documenting how &quot;inequality kills us all&quot;. He noted that the US was among the leaders in infant mortality and life expectancy in the 1950s.  Now the US is trailing most of the advanced industrial democracies per United Nations (2022).  He attributes the slow rate of improvement in public health in the US to increases in inequality.  That argument is less than perfect, because Figure 5 suggests that inequality in the US did not begin to increase until around 1975, but the divergence in public health between the US and other advanced industrial democracies seems more continuous between the 1950s and the present, 2023.  Beyond this, Graves and Samuelson (2022) noted that it is in everyone's best interest to help others with conditions that might be infectious to get competent medical assistance, because that reduces our risk of contracting their disease and possibly dying from it. Bezruchka (2023) cites documentation claiming that even the wealthy in the US have lower life expectancy than their counterparts in other advanced industrial democracies, because the high level of inequality in the US means that the ultra-wealthy in the US get exposed to more pathogens than their counterparts elsewhere.  See also Wilkinson and PIckett (2017).&lt;/ref&gt; To what extent might that increase in inequality be due to the structure of the major media?  

To what extent might you and I benefit from making it easier for millions of others to research different aspects of government policies including the &quot;adjustments&quot; in the US tax system embedded in the over 10 million words of US federal tax code and regulations documented with Figure 4 above, encouraging them to lobby the US Congress against the special favors granted to major political campaign contributors against the general welfare of everyone else? Everyone except the beneficiaries of such political corruption would likely benefit from the news that helps concerned citizens lobby effectively against such corruption, even if we did not participate in such citizen lobbying efforts and were completely ignorant of them.

== Media and war == 

:[[w:You've Got to Be Carefully Taught|''&quot;You've got to be taught to hate and fear. ... It's got to be drummed in your dear little ear.&quot;'']] 
:Lt. Cable in the [[w:South Pacific (musical)|1949 Rodgers and Hammerstein musical ''South Pacific'']].  

To what extent is it accurate to say that before anyone is killed in armed hostilities, the different parties to the conflict are polarized by the different media the different parties find credible?&lt;ref&gt;The role of the media in war has long been recognized.  It is commonly said that the first casualty of war is truth. Knightley (2004, p. vii) credits Senator Hiram Johnson as saying in 1917, &quot;The first casualty when war comes is truth.&quot; However, the Wikiquote article on &quot;[[Wikiquote:Hiram Johnson|Hiram Johnson]]&quot; says this quote has been, &quot;Widely attributed to Johnson, but without any confirmed citations of original source. ... [T]he first recorded use seems to be by Philip Snowden.&quot; (accessed 2023-07-22.)&lt;/ref&gt; 

This might seem obvious, but how can we quantify political polarization in a way (a) that correlates with the severity of the conflict and (b) can be used to evaluate the effectiveness of efforts to reduce the polarization? 

The [[w:International Panel on the Information Environment|International Panel on the Information Environment]] (IPIE) is a consortium of over 250 global experts developing tools to combat political polarization driven by the structure of the Internet.&lt;ref&gt;e.g., National Acadamies (2023).&lt;/ref&gt; The US Institute of Peace (2016) discusses &quot;Tools for Improving Media Interventions in Conflict Zones&quot;. Previous research in this area was summarized by Arsenault et al. (2011). One such tool might be video games.&lt;ref&gt;Caelin (2016).&lt;/ref&gt;

We suggest experimenting with interventions designed to reduce political polarization with some of the smallest but most intense conflicts: Interventions that require money could be more effectively tested with smaller, high intensity conflicts. With randomized controlled trials, it would be easier to measure a reduction from a higher-intensity conflict, and a smaller population could commit more money per capita with a relatively modest budget.

The [[w:Armed Conflict Location and Event Data Project|Armed Conflict Location and Event Data Project (ACLED)]] tracks politically relevant violent and nonviolent events by a range of state and non-state actors. Their data can help identify countries or geographic regions in conflict as candidates to be [[w:Randomized controlled trial|assigned randomly to experimental and control groups]], whose comparison can provide high quality data to help evaluate the impact of any intervention. Initial experiments of this nature might be done with a modest budget by working with organizations advocating nonviolence and with religious groups to recruit diaspora communities to do things recommended by experts in IPIE while also lobbying governments for funding. Any success can be leveraged into changes in foreign and military policies to make the world safer for all. 

Before discussing such experiments further, we consider a few case studies. 

=== Russo-Ukrainian War and the US Civil War === 

In the [[w:Russo-Ukrainian War|Russo-Ukrainian War]], Halimi and Rimbert (2023) describe &quot;Western media as cheerleaders for war&quot;. [[w:Joseph Stiglitz|Stiglitz]] (2002) noted this was a general phenomenon:  &quot;In periods of perceived conflict ... a combination of self-censorship and reader censorship may also undermine the ability of a supposedly free press to ensure democratic transparency and openness.&quot; Media organizations do not always do this solely to please their funders.  Reporters are killed&lt;ref&gt;Different lists of journalists killed for their work are maintained by the [[w:Committee to Protect Journalists|Committee to Protect Journalists]], (CPJ), [[w:Reporters Without Borders|Reporters without Borders]], and the [[w:International Federation of Journalists|International Federation of Journalists]].  CPJ has claimed that their numbers are typically lower, because their confirmation process may be more rigorous.  See Committee to Protect Journalists (undated) and the Wikipedia articles on &quot;[[w:Committee to Protect Journalists|Committee to Protect Journalists]]&quot;, &quot;[[w:Reporters Without Borders|Reporters without Borders]]&quot;, and the &quot;[[w:International Federation of Journalists|International Federation of Journalists]]&quot;, accessed 2023-07-11.&lt;/ref&gt; or jailed and news outlets closed to prevent them from disseminating information that people with power do not want distributed.  Early in the Civil War in the US (1861-1865), some newspapers in the North said the US should let the South secede, because that would be preferable to war.  Angry mobs destroyed some offices and printing presses.  One editor &quot;was forcibly taken from his house by an excited mob, ... covered with a coat of tar and feathers, and ridden on a rail through the town.&quot; Others changed their policies &quot;voluntarily&quot;, recognizing threats to their lives or property or to a loss of audience.&lt;ref&gt;Harris (1999, esp. p. 100).&lt;/ref&gt;

=== Hitler ===

Fulda (2009) studied the co-evolution of newspapers and party politics in Germany, focusing primarily on Berlin, 1924-1933. During that period, the [[w:Nazi Party|Nazis (NSDAP, National Socialist German Workers' Party, Nationalsozialistische Deutsche Arbeiterpartei)]] grew from 2.6 percent of the votes for the [[w:Reichstag (Nazi Germany)|Reichstag (German parliament)]] in 1928 to 44 percent in 1933. Fulda described exaggerations in the largely tabloid press of an indecisive government incapable of managing either the economy or the increasing political violence, blamed excessively on Communists, and the potential for civil war. This turned the Nazis into an attractive choice for voters desperate for decisive action.&lt;ref&gt;Fulda (2009, Abstract, ch. 6, &quot;War of Words: The Spectre of Civil War, 1931–2&quot;.&lt;/ref&gt; 

After the 1933 elections, the Reichstag passed the &quot;[[w:Enabling Act of 1933|Enabling Act of 1933]], which gave Hitler's cabinet the right to enact laws without the consent of parliament. The Nazis then began full censorship of the newspapers, physically beating, imprisoning and in some cases killing journalists, as the leading publishers acquiesced. The primary sources of news during that period was newspapers; German radio was relatively new during that period and carried very little news. Most newspapers were [[w:Tabloid journalism|tabloids]], interested in either making money or promoting a party line with minimal regard for fact checking. A big loser in this was the right‐wing press magnate [[w:Alfred Hugenberg|Alfred Hugenberg]], whose political mismanagement led to the substantial demise of his [[w:German National People's Party|German National People's Party (Deutschnationale Volkspartei, DNVP)]], mostly benefitting Hitler.&lt;ref&gt;Fulda (2009).&lt;/ref&gt; 

This suggests the need for a [[w:Counterfactual history|counterfactual analysis of this period]], asking what kinds of changes in the structure of the media ecology might have prevented the rise of the Nazis? In particular, to what extent might a more diverse local news environment supported by citizen-directed subsidies as suggested herein have reduced the risk of a demise of democracy? And might some sort of [[w:Fairness Doctrine|fairness doctrine]] have helped?&lt;ref name=fairness&gt;Wikipedia &quot;[[w:FCC fairness doctrine|FCC fairness doctrine]]&quot;, accessed 2023-07-21.&lt;/ref&gt; And how might different rules for distributing different levels of funding to local news outlets impact the level of democratization? (Threats to democracy include legislation like the German Enabling Act of 1933 and other situations that allow an executive to successfully ignore the will of an otherwise democratic legislature, a [[w:self-coup|self-coup]], as well as a military coup.)

=== Stalin and Putin === 

[[File:Russian economic history 1885-2018.svg|thumb|Figure 7. Gross Domestic Product per person in Russian 1885-2018 in thousands of 2011 dollars]]

A 2017 survey asking Russians to name 10 of the world’s most prominent personalities listed Joe Stalin and Vladimir Putin as the top two with 38 and 34 percent, respectively. When the study was redone in 2021, Putin had slipped from number 2 to number 5. Stalin still led with 39 percent followed by Vladimir Lenin with 30 percent, Poet Alexander Pushkin and tsar Peter the Great with 23 and 19 percent each, then Putin with 15 percent.&lt;ref&gt;&lt;!-- Putin Plummets, Stalin Stays on Top in Russians’ Ranking of ‘Notable’ Historical Figures – Poll--&gt;{{cite Q|Q123197680}} &lt;!-- The most outstanding personalities in history (in Russian) --&gt;{{cite Q|Q123197317}}&lt;/ref&gt; 

It may be difficult for some people in the West to understanding how Stalin and Putin could be so popular, given the way they have been typically described in the mainstream Western media.&lt;ref&gt;[[w:Joseph Stalin|Joseph Stalin]] got much better press in the US during the Great Depression and World War II than he has gotten since 1945.&lt;/ref&gt; However, this is relatively easy to understand just by looking at the accompanying plot (Figure 7) of average annual income in that part of the world between 1885 and 2018: Both Stalin and Putin inherited economies that had fallen dramatically in the previous years and supervised dramatic improvements. Putin's decline between 2017 and 2021 may also be understood from this plot, because it shows how the dramatic growth that began around the time that Putin became acting President of Russia has slowed substantially since 2012. 

Similar comments could be made about the Vietnam war and the &quot;War on Terror&quot;.&lt;ref&gt;The Wikiversity article on &quot;[[Winning the War on Terror]]&quot; discusses the role of the media in the &quot;War on Terror&quot; and other conflicts including Vietnam.&lt;/ref&gt; To what extent can the experiments described in this article contribute to understanding the role of the media in stoking hate and fear, and how that might be impacted by citizen-direct subsidies for more and more diverse local media?

=== Iraq and the Islamic State ===

[[w:Fall of Mosul|In 2014 in Mosul, two Iraqi army divisions totaling 30,000 and another 30,000 federal police]] were overwhelmed in six days by roughly 1,500 committed Jihadists. Four months later, ''Reuters'' reported that, &quot;there were supposed to be close to 25,000 soldiers and police in the city; the reality ... was at best 10,000.&quot; Many of the missing 15,000 were &quot;ghost soldiers&quot; kicking back half their salaries to their officers. Also, &quot;[i]nfantry, armor and tanks had been shifted to Anbar, where more than 6,000 soldiers had been killed and another 12,000 had deserted.&quot;&lt;ref&gt;Parker et al. (2014).&lt;/ref&gt; 

To what extent might the political corruption and low moral documented in that ''Reuters'' report have been allowed to grow to that magnitude if Iraq had had a vigorous adversarial press, as discussed in this article? Instead, Paul Bremer, who was appointed as the [[w:Paul Bremer#Provisional coalition administrator of Iraq|Provisional coalition administrator of Iraq]] just over a week after President George W. Bush's [[w:Mission Accomplished speech|Mission Accomplished speech]] of 2003-05-01, imposed strict press censorship.&lt;ref&gt;McChesney and Nichols (2010, p. 242).&lt;/ref&gt; McChesney and Nichols contrasted this with General Eisenhower, who &quot;called in German reporters [after the official surrender of Nazi Germany in WW II] and told them he wanted a free press. If he made decisions that they disagreed with, he wanted them to say so in print.&quot;&lt;ref&gt;McChesney and Nichols (2010, Appendix II. Ike, MacArthur and the Forging of Free and Independent Press, pp. 241-254).&lt;/ref&gt;

=== Israel-Palestine ===

::''Those who make peaceful revolution impossible will make violent revolution inevitable.''
:::-- John F. Kennedy (1962)

To what extent is the [[w:Israeli–Palestinian conflict|Israeli–Palestinian conflict]] driven by differences in the media consumed by the different parties to that conflict? 

* To what extent are the supporters of Israel aware of violent acts committed by Palestinians but are ''unaware'' of the actions by Israelis that have motivated those violent acts? 
* Similarly, to what extent are the supporters of Palestinians unaware of or downplay the extent to which violence by Palestinians motivate the actions of Israel against them? 

To what extent are these differences in perceptions between supporters of Israel and supporters of Palestinians driven by differences in the media each find credible? 

What can be done to bridge this gap?  

[[w:Gene Sharp|Gene Sharp]], [[w:Mubarak Awad|Mubarak Awad]], and other supporters of [[w:nonviolence|nonviolence]] have suggested that when nonviolent direct action works, it does so by exposing a gap between the rhetoric [supported by the major media] and the reality of their opposition. Over time, this gap erodes pillars of support of the opposition. One example was the nonviolence of the [[w:First Intifada|First Intifada]] (1987-1993), which were protests against &quot;beatings, shootings, killings, house demolitions, uprooting of trees, deportations, extended imprisonments, and detentions without trial.&quot;&lt;ref&gt;Ackerman and DuVall (2000, p. 407).&lt;/ref&gt; As that campaign began, Israel got so much negative press for killing nonviolent protestors, that Israeli Defense Minister [[w:Yitzhak Rabin|Yitzhak Rabin]] ordered his soldiers NOT to kill but instead to shoot to wound. As the negative press continued, he issued wooden and metal clubs with orders to break bones.&lt;ref&gt;Shlaim (2014).&lt;/ref&gt; As the negative press still continued, Rabin ran for Prime Minister on a platform of negotiating with Palestinians. His victory and subsequent negotiations led to the [[w:Oslo Accords|Oslo Accords]] and the joint recognition of each other by the states of Israel and [[w:State of Palestine|Palestine]]. The West Bank and Gaza have continued under Israeli occupation since with some services provided by the official government of Palestine. 

During the Intifada, Israel tried to infiltrate the protestors with [[w:agent provocateur|agents provocateurs]] in Palestinian garb. They were exposed and neutralized until Israel deported 481 people they thought were leading the nonviolence who were accepted in other countries and imprisoned tens of thousands of others suspected of organizing the nonviolence. Finally, they got the violence needed to justify a massively violent repression of the Intifada.&lt;ref&gt;King (2007).&lt;/ref&gt;

The general thrust of this current analysis suggests a two pronged intervention to reduce the risk of a continuation of the violence that has marked Israel-Palestine since at least 1948: 
# Offer nonviolence training to all Palestininans, Israelis and supporters of either interested in the topic. This is the opposite of the policies Israel pursued during the First Intifada, at least according to the references cited in this discussion of that campaign.&lt;ref&gt;It also is the opposite of the decision of the US Supreme Court in ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', which ruled that teaching nonviolence to someone designated as a terrorist was a crime under the [[w:Patriot Act|Patriot Act]], as it provided &quot;material support to&quot; a foreign terrorist organization.&lt;/ref&gt;
# Provide citizen-direct subsidies to local news nonprofits in the West Bank and Gaza at, e.g., 0.15 percent of GDP, as recommended by McChesney and Nichols, cited above. 

How can we evaluate the budget required for such an experiment? The nominal GDP of the [[w:State of Palestine|State of Palestine]] in 2021 was estimated at $18 billion; 0.15% of that is $27 million. Add 10% for research to get $30 million per year. That ''annual'' cost for the media component of this proposed intervention is 12% of the billion Israeli sheckels ($246 million) that the Gaza war was costing Israel ''each day'' in the early days of the [[w:Israel-Hamas war|Israel-Hamas war]], according to the Israeli Finance Minister on 2023-10-25.&lt;ref&gt;Reuters (2023-10-25).&lt;/ref&gt; As this is being written, that war has continued for over 100 days. If the average daily cost of that war to Israel during that period has been $246 million, then that war will have already cost Israel over $24.6 billion. And that does not count the loss of lives and the destruction of property in Gaza and the West Bank. 

How much would training in nonviolence cost? That question would require more research, but if it were effective, the budget would seem to be quite modest compared to the cost of war, even if it were several times the budget for citizen-directed subsidies for local news in Palestine as just suggested.

== The decline of newspapers == 

[[File:Newspapers as a percent of US GDP.svg|thumb|Figure 8.  US newspaper revenue 1955-2020 as a percent of GDP.&lt;ref&gt;&quot;USnewspapers&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

McChesney and Nichols (2022) noted that US newspaper revenue as a percent of GDP fell from over 1 percent in 1956 to less than 0.1 percent in 2020;  see Figure 8.  Abernathy (2020) noted that the US lost more than half of all newspaper journalists between 2008 and 2018.&lt;ref&gt;Abernathy (2020, p. 22).&lt;/ref&gt; A quarter of US newspapers closed between 2004 and 2020,&lt;ref&gt;Abernathy (2020, p. 21).&lt;/ref&gt; and many that still survive are publishing less, creating &quot;news deserts&quot; and &quot;ghost newspapers&quot;, some with no local journalists on staff.&lt;ref&gt;Abernathy (2020) documented the problem of increasing &quot;news deserts and ghost newspapers&quot; in the US. A local jurisdiction without a local news outlet has been called a &quot;news desert&quot;.  She uses the term &quot;ghost newspapers&quot; to describe outlets &quot;with depleted newsrooms that are only a shadow of their former selves.&quot;  Some “ghost newspapers” continue to publish with zero local journalists, produced by reporters and editors that don't live there. One example is the ''Salinas Californian'', a 125-year-old newspaper in Salinas, California, which lost its last paid journalist 2022-12, according to the Los Angeles Times (2023-03-27).  They continue to publish, though &quot;The only original content from Salinas comes in the form of paid obituaries, making death virtually the only sign of life at an institution once considered a must-read by many Salinans.&quot; A leading profiteer in this downward spiral is reportedly [[w:hedge fund|hedge fund]] [[w:Alden Global Capital|Alden Global Capital]].  Threisman (2021) reported that, &quot;When this hedge fund buys local newspapers, democracy suffers&quot;.  And Benton (2021) said, &quot;The vulture is hungry again: Alden Global Capital wants to buy a few hundred more newspapers&quot;. Hightower (2023) describes two organizations fighting this trend.  One is National Trust for Local News, a nonprofit that recently bought several local papers and &quot;is turning each publication over to local non-profit owners and helping them find ways to become sustainable.&quot; The other is CherryRoad Media, which &quot;bought 77 rural papers in 17 states, most from the predatory Gannett conglomerate that wanted to dump them&quot;, and is working to &quot;return editorial decision-making to local people and journalists ... and ... reinvest profits in real local journalism that advances democracy.&quot; News outlets acquired by something like the National Trust for Local News should be eligible for citizen-directed subsidies for local news, as discussed below, after their ownership was officially transferred to local humans.  Outlets acquired by organizations like CheeryRoad Media would not be eligible as long as they remained subsidiaries.&lt;/ref&gt; More recent news continues to be dire. The Fall 2023 issue of ''Columbia Journalism Review'' reported that 2023 &quot;has become media’s worst year on record for job losses&quot;.&lt;ref&gt;Columbia Journalism Review (2023).&lt;/ref&gt; Substantial advertising revenue has shifted to the &quot;click economy&quot;, where advertisers pay for clicks, especially on social media.&lt;ref&gt;Carter (2021).&lt;/ref&gt;

Newpapers in other parts of the world have also experienced substantial declines in revenue. In 2013 German law was changed to inclued &quot;[[w:Ancillary copyright for press publishers|Ancillary copyright for press publishers]]&quot;, also called a &quot;link tax&quot;. However, this law was declared invalid in 2019 the European Court of Justice (ECJ), because it had been submitted in advance to the [[w:European Commission|EU Commission]], as required.&lt;ref&gt;&lt;!--Axel Kannenberg (2019) ECJ: German ancillary copyright law invalid for publishers, heise online--&gt;{{cite Q|Q124051681|title= ECJ: German ancillary copyright law invalid for publishers}}&lt;/ref&gt; Before that ECJ decision, Google had removed newspapers from Google News in Germany. German publishers then reached an agreement with Google after traffic to their websites plummeted.&lt;ref&gt;&lt;!--Dominic Rushe (2014) Google News Spain to close in response to story links 'tax', Guardian--&gt;{{cite Q|Q124051847}}&lt;/ref&gt; Building on that and similar experience in Spain, the European Union adopted a [[w:Directive on Copyright in the Digital Single Market|Directive on Copyright in the Digital Single Market]] in 2019. A similar link tax proposal in Canada led [[w:Meta|Meta]], the parent company of Facebook, to withdraw news from Canada, and Google agreed to 'pay about $100 million a year into a new fund to support “news&quot;' in Canada. As of 2023-11-30, California was still considering a link tax.&lt;ref&gt;&lt;!--Ken Doctor (2023) Forget the link tax. Focus on one key metric to “save local news, NiemanLab--&gt;{{cite Q|Q124051930}}&lt;/ref&gt;

== Threats from social media == 

The growth of social media has been wonderful and terrible.  It has been wonderful in making it easier for people to maintain friendships and family ties across distances.&lt;ref&gt;Friedland (2017) noted that the Internet works well at the global level, helping people get information from any place in the world, and at the micro level, e.g., with Facebook helping people with similar diseases find one another.  It does not work well at the '“meso level arenas of communication” in the middle.  They're not big enough to aggregate all the scale that goes into creating a worldwide web or even a Wikipedia. See also Lloyd and Friedland (2016).&lt;/ref&gt; 

But it has also been terrible as  &quot;antisocial media&quot;&lt;ref&gt;Vaidhyanathan (2018).&lt;/ref&gt; have been implicated in the relatively recent rise in dysfunctional and counterproductive political polarization and violence. Ding et al. (2023) document, &quot;Same words, different meanings&quot; in their use by [[w:CNN|CNN]] and [[w:Fox News|Fox News]] and how that has interacted with word usage on Twitter between 2010 and 2020 to increase political polarization, &quot;impeding rather than supporting online democratic discourse.&quot;&lt;ref&gt;See also Ashburn.&lt;/ref&gt; 

Extreme examples of this increase have included violent efforts to prevent peaceful transitions of power in the US&lt;ref&gt;Wikipedia &quot;[[w:January 6 United States Capitol attack|January 6 United States Capitol attack]]&quot;, accessed 2023-05-09.&lt;/ref&gt; and Brazil.&lt;ref&gt;Wikipedia &quot;[[w:2023 Brazilian Congress attack|2023 Brazilian Congress attack]]&quot;, accessed 2023-05-09.&lt;/ref&gt; These changes even threaten the national security of the US and its allies,&lt;ref&gt;McMaster (2020). Zuboff (2019) noted that data on many aspects of ordinary daily life are captured and used by people with power for various purposes.  For example, data on people's locations captured from their mobile phones are used to try to sell them goods and services.  Data on a child playing with a smart Barbie doll are used to inculcate shopping habits in child and caregiver.  If you are late on a car payment, your keys can be deactivated until a tow truck can arrive to haul it away.  To what extent do the major media today have conflicts of interest in honestly reporting on this?  How might the experiments proposed herein impact the commercial calculus of major media and the political economy more generally?&lt;/ref&gt; according to [[w:H. R. McMaster|H. R. McMaster]],&lt;ref&gt;Wikipedia &quot;[[w:H. R. McMaster|H. R. McMaster]]&quot;, accessed 2023-05-09.&lt;/ref&gt; President Trump's second national security advisor. 

Various responses to these concerns have been suggested, beyond the recommendations of McChensey and Nichols.  These include the following:&lt;ref&gt;See also the section on &quot;&quot;[[International Conflict Observatory#Suggested responses to these concerns|Suggested reponses to these concerns]]&quot; in the Wikiversity article on &quot;[[International Conflict Observatory]]&quot;.&lt;/ref&gt;:

* Make internet companies liable for defamation in advertisements, similar to print media and broadcasting.&lt;ref&gt;See Baker (2020) and the Wikiversity article on &quot;[[Dean Baker on unrigging the media and the economy]], accessed 2023-07-26.&lt;/ref&gt;  
* Tax advertising revenue received by large internet companies and use that to fund more local media.&lt;ref&gt;Karr and Aaron (2019).&lt;/ref&gt; 
* Replace advertising as the source of funding for social media with subscriptions.&lt;ref&gt;Frank (2021) wrote, &quot;[D]igital aggregators like Facebook ... make money not by charging for access to content but by displaying it with finely targeted ads based on the specific types of things people have already chosen to view. If the conscious intent were to undermine social and political stability, this business model could hardly be a more effective weapon. ... [P]olicymakers’ traditional hands-off posture is no longer defensible.&quot;&lt;/ref&gt; 

To these suggestions, we add the following:  

* Allow some of but not all citizen-directed subsidies for news to go to social media outlets, as suggested below.  
* Require that all organizations whose income depends on promoting or &quot;boosting&quot; content, whether in advertisements or &quot;underwriting spots&quot; or [[w:clickbait|clickbait]], to provide copies of the ads, underwriting spots and clickbait to a central repository like the [[w:Internet Archive|Internet Archive]].  
* Use advertising to discuss overconfidence and encourage people to talk politics with humility and respect, recognizing that the primary differences they have with others are the media they consume.&lt;ref&gt;For studies of ad campaigns in other contexts, see Piwowarski et al. (2019) and Tom-Yov (2018), cited above in discussing &quot;Reducing political polarization&quot;.&lt;/ref&gt;

== How to counter political polarization ==

More research seems to be needed on how to counter the relatively recent increases in political polarization. For example, might some form of [[w:Fairness doctrine|fairness doctrine]]&lt;ref name=fairness/&gt; help reduce political polarization? [[w:Fairness doctrine#Opposition|Conservative leaders are vehemently opposed]], insisting it would be an attack on First Amendment rights. However, as noted above, the tabloid media of Germany seems to have contributed to Hitler's rise to power between 1924 and 1933. How is the increase in political polarization since 1987 and 2004 different from the disregard for fairness of the news media that helped bring Hitler to power? One example: The lawsuit ''[[w:Dominion Voting Systems v. Fox News Network|Dominion Voting Systems v. Fox News Network]]'' was settled with Fox agreeing to to pay Dominion $787.5 million while acknowledging that Fox had knowingly and intentionally made false and defamatory statements about Dominion to avoid losing audience to media outlets that continued to claim fraudulently that Donald Trump not Joe Biden had won the 2020 US presidential election. The settlement permitted Fox to avoid apologizing publicly, which could have threatened their audience share. That settlment was less than 6 percent of Fox's 2022 revenue of $14 billion.&lt;ref&gt;&lt;!--Fox earnings release for the quarter and fiscal year ended June 30, 2022--&gt;{{cite Q|Q124003735}}&lt;/ref&gt; Evidently, ''if that decision made a difference of 6 percent in their audience ratings, Fox made money from defaming Dominion even after paying them $787.5 million.'' If so, it was a good business decision, especially since they did not have to publicly apologize.  

To what extent did Fox's lies about Dominion contribute to the [[w:January 6 United States Capitol attack|mob attacks on the US Capitol on January 6, 2021]], trying to prevent the US Congress from officially declaring that Joe Biden had won the 2020 elections? And what are elected officials prepared to do to improve understanding of what contributes to increases in political polarization and how political differences can be made less lethal and more productive?

== McChesney and Nichols' Local Journalism Initiative ==

As noted above, McChesney and Nichols (2021, 2022) propose a &quot;Local Journalism Initiative&quot;, distributing 0.15 percent of GDP to local news nonprofits via local elections.  They based this partly on their earlier work suggesting that subsidies for newspapers in the US in 1840 was around 0.2 percent of GDP.&lt;ref&gt;McChesney and Nichols (2010, 2016).&lt;/ref&gt;

=== McChesney and Nichols' eligibility criteria === 

To be eligible, McChesney and Nichols say the recipient of such funds should satisfy the following:&lt;ref&gt;McChesney and Nichols (2021, 2022).  They also suggest having the US Postal Service administer this with elections every three years.&lt;/ref&gt; 
* Be a local nonprofit with at least six months of history, so voters could know their work.  
* Be locally based with at most 75 percent of salaries going to local residents.  
* Be completely independent, not a subsidiary of a larger organization.
* Produce and publish original material at least five days per week on their website for free, explicitly in the public domain. 
* Each voter is asked to vote for at least three different local news outlets to support diversity. 
* No single news outlet should get more than 25 percent of that jurisdiction's annual budget for local news subsidies.  
* Each recipient of these subsidies should get at least 1 percent of the vote to qualify, or 0.5 percent of the vote in political jurisdictions with over 1 million people. Diversity and competition are crucial.  
* There will be no content monitoring:  Government bureaucrats will not be allowed to decide what is &quot;good journalism&quot;.  That's up to the voters.  
* Voting would be limited to those 18 years and older.  

=== Alternatives === 

Some aspects of this might be relaxed for at least some political jurisdictions included in an experiment.  For example, might it be appropriate to allow for-profit news outlets to compete for these subsidies as long as they meet the other criteria?&lt;ref&gt;Kaiser (2021) noted that nonprofits in the US cannot endorse political candidates and are limited in how they can get involved in debates on political issues.  Do restrictions like these contribute to the general welfare?  Or might the public interest be better served with citizen-directed subsidies for media that might be more partisan?  This is one more question that might be answered by appropriate experimentation.&lt;/ref&gt; However, we prefer to retain the rules requiring recipients to be local and completely independent, at least for many experimental jurisdictions.&lt;ref&gt;Various contributors to Islam et al., eds. (2002) raised questions about concentrations of power in large media organizations, especially Herman, ch. 4, pp. 61-81 (73-97/336 in pdf).  Djankov et al. (2002) found that &quot;Government ownership of the media is detrimental to economic, political, and-most strikingly-social outcomes&quot;, including education and health.&lt;/ref&gt; If citizen-directed subsidies for local news go to for-profit organizations, to what extent should their finances be transparent, e.g., otherwise complying with the rules for 501(c)(3)s? Might it also be appropriate to allow some portion of these funds to be distributed to noncommercial ''social media'' outlets that submitted all their content to a public, searchable database like the [[w:Internet Archive|Internet Archive]]?   

News written by people paid with these subsidies should be available under a free license like Creative Commons Attribution-ShareAlike (CC BY-SA) 4.0 International license but not necessarily in the public domain:  Other media outlets should be free to further disseminate the news while giving credit to the organization that produced it.   

Many countries have some form of [[w:community radio|community radio]].  Some of those radio stations include what they call news and / or public affairs, and some of those are made available as podcasts via the Internet.&lt;ref&gt;In the US, many of these stations collaborate via organizations such as the [[w:National Federation of Community Broadcasters|National Federation of Community Broadcasters]], the [[w:List of Pacifica Radio stations and affiliates#Radio Stations#Affiliates|Pacifica Network Affiliates]], and the [[w:Grassroots Radio Coalition|Grassroots Radio Coalition]]. One such station with regular local news produced by volunteers in [[w:KBOO|KBOO]] in [[w:Portland, Oregon|Portland, Oregon]]; see Loving (2019).&lt;/ref&gt; If their &quot;news &amp; public affairs&quot; programs are subsequently posted to a website as podcasts, preferably accompanied by some text if not complete transcripts, under a license no more restrictive than CC BY-SA, that should make them eligible for subsidies under the criteria mentioned above if they add at least one new podcast of that nature five days per week.  If the programming of this nature that they produce is ''not'' available on the web or under an appropriate license, part of any experiments as discussed here might include offers to help such radio stations become eligible.  

Might it be wise to allow children to vote for news organizations they like?  Ryan Sorrell, founder and publisher of the Kansas City Defender, insists that, &quot;young people ... are very interested in news. It just has to be produced and packaged the right way for them to be interested in consuming it&quot;.&lt;ref&gt;Holmes (2022).&lt;/ref&gt; The French-language [[:fr:w:Topo (revue)|''Topo'']] present news and complex issues in comic strip format. Their co-editor in chief insists, &quot;there are plenty of ways to get young people interested in current affairs&quot;.&lt;ref&gt;Biehlmann (2023).&lt;/ref&gt; Might allowing children to vote for news outlets they like increase public interest in learning and in civic engagement among both children and their caregivers?  Should this be tested in some experimental jurisdictions?&lt;ref&gt;We may not want infants who cannot read a simple children's book to vote for &quot;news&quot;, but if they can read the names of eligible local news outlets on a ballot, why not encourage them to vote?  As Rourmeen Islam wrote in 2002, &quot;erring on the side of more freedom rather than less would appear to cause less harm.&quot; (World Bank, 2002, pp. 21-22; 33-34/336 in pdf).&lt;/ref&gt;

Some of the money may go to media outlets that seem wacko to many voters.  However, how different might that be from the current situation? Most importantly, if these subsidies have the effect that Tocqueville reported from 1831, they should be good for democracy and for broadly shared peace and prosperity for the long term:  They could stimulate public debate, and wacko media might have ''less'' power than they currently do, with &quot;each separate journal exercis[ing] but little authority; but the power of the periodical press [being] second only to that of the people.&quot;&lt;ref&gt;Tocqueville (1835; 2001, p. 94).&lt;/ref&gt; 

Tocqueville's comparison of newspapers in France and the US in 1831 is echoed in Cagé's (2022) concern about &quot;the Fox News effect&quot; in the US and that of Bolloré in France.  She cites research claiming that biases in Fox News made major contributions to electing Republicans in the US since 2000.&lt;ref&gt;Cagé (2022, pp. 21-22, 59-60).  She cited DellaVigna and Ethan Kaplan (2007), who reported that Fox News had introduced cable programming into 20 percent of town in the US between 1996 and 2000.  They found that the presence of Fox increased the vote share for Republicans between 0.4 and 0.7 percentage points over neighboring non-Fox towns that seemed otherwise indistinguishable.  In 2000 Fox News was available in roughly 35 percent of households, which suggests that Fox News shifted the nationwide vote tally by between 0.15 and 0.2 percentage points.  They conclude that this shift was small but likely decisive in the close 2000 US presidential election.&lt;/ref&gt; These shifts, including changes by the conservative-leaning broadcasting company, Sinclair Broadcast Group, reportedly made a substantive contribution to the election of [[w:Donald Trump|Donald Trump]] as US President in 2016, while a comparable estimate of the impact of changes in MSNBC &quot;is an imprecise zero.&quot;&lt;ref&gt;Cagé (2022, pp. 21-22).  Miho (2022) analyzes the timing of the introduction of biased programming by the conservative-leaning broadcasting company, Sinclair Broadcast Group, between 1992 and 2020, comparing counties in the US with and without a Sinclair station.  This work estimates a 2.5 percentage point increase in the Republican vote share during the 2012 US presidential election and double that during the 2016 and 2020 presidential elections with comparable increases in Republican representation in the US Congress.&lt;/ref&gt; In France, she provides documentation claiming that the media empire of French billionaire [[w:Vincent Bolloré|Vincent Bolloré]] has made a major contribution to the rise of far-right politician [[w:Éric Zemmour|Éric Zemmour]] and is buying media in Spain.&lt;ref&gt;Cagé (2022, pp. 24, 60)&lt;/ref&gt; The pattern is simple:  Fire journalists and replace them with talk shows, which are cheaper to produce and are popular, evidently exploiting the [[w:overconfidence effect|overconfidence effect]].  

To what extent is the increase in political polarization since 1987&lt;ref name=fairness/&gt; and 2004&lt;ref&gt;Wikipedia &quot;[[w:Facebook|Facebook]]&quot;, accessed 2023-07-21.&lt;/ref&gt; due to increased concentration of ownership of both traditional and social media (and how those organizations make money selling changes in audience behaviors to the people who give them money)? To the extent that this increase in polarization has been driven by those changes in the media, citizen-directed subsidies for diverse news should reverse that trend. This hypothesis can be tested by experiments like those proposed herein.

== Roadmap for local news == 

Green et al. (2023) describe &quot;an emerging approach to meeting civic information needs&quot; in a &quot;Roadmap for local news&quot;. This report insists that society needs &quot;civic information&quot;, not merely &quot;news&quot;. It summarizes interviews with 51 leaders from nonprofit and commercial media across all forms of distribution (print, radio, broadcast, digital, SMS) in member organizations, news networks, news funders and researchers. They say that, &quot;Rampant disinformation is being weaponized by extremists&quot;, and &quot;Democratic participation and representation are under threat.&quot; They recommend four strategies to address &quot;this escalating information crisis&quot;: 

# Coordinate work around the goal of expanding “civic information,” not saving the news business;
# Directly invest in the production of civic information;
# Invest in shared services to sustain new and emerging civic information networks; and
# Cultivate and pass public policies that support the expansion of civic information while maintaining editorial independence. 

Part of the motivation for this article on &quot;Information is a public good&quot; is the belief that solid research on the value of such interventions should both (a) make it easier to get the funding needed, and (b) help direct the funding to interventions that seem to make the maximum contributions to improving broadly shared peace and prosperity for the long term at minimum cost.

== Budgets for experiments == 

What factors should be considered in evaluating budgets for experiments to estimate the impact of citizen-directed subsidies for news?   

[[File:Advertising as a percent of Gross Domestic Product in the United States.svg|thumb|Figure 9.  Advertising as a percent of Gross Domestic Product in the United States, 1919 to 2007.&lt;ref name=ads&gt;Galbi (2008).&lt;/ref&gt;]]  

Rolnik et al. (2019) suggested that $50 per person, roughly 0.08 percent of US GDP, might be enough.  However, that's a pittance compared to the revenue lost by newspapers in the US since 1955, as documented in Figure 8 above.  It's also a pittance compared to the money spent on advertising (see Figure 9):  Can we really expect local media funded with only 0.08 or 0.15 percent of GDP to compete with media funded by 2 percent of GDP?  Maybe, but that's far from obvious.  Might it be prudent to fund local journalism in some experimental jurisdictions at levels exceeding the money spent on advertising, i.e., at roughly 2 percent of GDP or more?  If information is a public good, as suggested by the research summarized here, then such high subsidies would be needed in some experimental jurisdictions, because the maximum of anything (including net benefits = benefits minus costs) cannot be confidently identified without conducting some experiments ''beyond the point of diminishing returns''.&lt;ref&gt;A parabola can be estimated from three distinct points.  However, in fitting a parabola or any other mathematical model to empirical data, one can never know if an empirical phenomenon has been adequately modeled and a maximum adequately estimated without data near the maximum and on both sides of it (unless the maximum is at a boundary, e.g., 0).  See, e.g., Box and Draper (2007).&lt;/ref&gt;  

[[File:AccountantsAuditorsUS.svg|thumb|Figure 10.  Accountants and auditors as a percent of the US workforce.&lt;ref name=actg&gt;Accountants and auditors as a percent of US households, 1850 - 2016, using the OCC1950 occupation codes in a sample of households available from from the [[w:IPUMS|Integrated Public Use Microdata Series at the University of Minnesota (IPUMS)]].  For more detail see the &quot;AccountantsAuditorsPct&quot; data set in the &quot;Ecdat&quot; package and the &quot;AccountantsAuditorsPct&quot; vignette in the &quot;Ecfun&quot; package available from within the [[w:R (programming language)|R (programming language)]] using 'install.packages(c(&quot;Ecdat&quot;, &quot;Ecfun&quot;), repos=&quot;http://R-Forge.R-project.org&quot;)'.&lt;/ref&gt;]] 

Also, news might serve a roughly comparable function to accounting and auditing, as both help reduce losses due to incompetence, malfeasance and fraud.  Two points on this:  

# CONTROL FRAUDS:  Black (2013) noted that many heads of organizations can find accountants and auditors willing to certify accounting reports they know to be fraudulent. Black calls such executives &quot;control frauds.&quot;&lt;ref&gt;Black (2013). &lt;/ref&gt; Primary protections against these kinds of problems are vigorous, independent journalists and more money spent on independent evaluations beyond the control of such executives.  In this regard, we note two major differences between the [[w:Savings and loan crisis|Savings &amp; Loan scandal]] of the late 1980s and early 1990s&lt;ref&gt;Wikipedia &quot;[[w:Savings and loan crisis|Savings and loan crisis]]&quot;, accessed 2023-06-25.&lt;/ref&gt; and the [[w:2007–008 financial crisis|international financial meltdown of 2007-2008]]:&lt;ref&gt;Wikipedia &quot;[[w:2007–008 financial crisis|2007–008 financial crisis]]&quot;, accessed 2023-06-25.&lt;/ref&gt; First the major banks by 2007 were much bigger and controlled much larger advertising budgets than the Saving &amp; Loan industry did 15-20 years earlier.  This meant that major media had a much bigger conflict of interest in honestly reporting on questionable activities of these major accounts.  Second, the major banks had made much larger political campaign contributions to much larger portions of both the US House and Senate. However, might the massive amounts of big money spent on campaign finance have been as effective if the major media did not have such a conflict of interest in exposing more details of the corrosive impact of major campaign donors on the quality of government?  To what extent might this corrosive impact be quantified in experimental polities?  
# ADEQUATE RESEARCH OF OUTCOMES:  Many nonprofits and governmental agencies officially have outcome measures, but many of those measures tend to be relatively superficial like the number of people served.  It's much harder to evaluate the actual benefits to the people served and to society.  For example, the Perry Preschool&lt;ref&gt;Schweinhart et al. (2005).  See also Wikipedia &quot;[[w:HighScope|HighScope]]&quot;, accessed 2023-06-15. &lt;/ref&gt; and Abecedarian&lt;ref&gt;e.g., Sparling and Meunier (2019).  See also Wikipedia &quot;Abecedarian Early Intervention Project&quot;, accessed 2023-06-25.&lt;/ref&gt; programs divided poor children and caregivers into experimental and control groups and followed them for decades to establish that their interventions were enormously effective.&lt;ref&gt;For more recent research on the economic value of high quality programs for early childhood development, see, e.g., &lt;!-- &quot;The Heckman Equation&quot; website (heckmanequation.org)--&gt;{{cite Q|Q121010808}}, accessed 2023-07-29.&lt;/ref&gt; Meanwhile, US President Lyndon Johnson's [[w:Great Society|Great Society]] programs,&lt;ref&gt;Wikipedia &quot;[[w:Great Society|Great Society]]&quot;, accessed 2023-07-11.&lt;/ref&gt; and [[w:Head Start|Head Start]] in particular, did not invest as heavily in research.  That lack of documentation of results made them a relatively easy target for political opponents claiming that government is the problem, not the solution. These counter arguments were popularized by US President Ronald Reagan and UK Prime Minister Margaret Thatcher to justify reducing or eliminating government funds for many such programs. Banerjee and Duflo (2019) summarized relevant research in this area by saying that the programs were not the disaster that Reagan, Thatcher, and others claimed, but they were also not as efficient and effective as they could have been, because many local implementations were underfunded, poorly managed and poorly evaluated.   Bedasso (2021) analyzed World Bank projects completed from 2009 to 2020, concluding that high quality monitoring and evaluation on average made a major contribution to the positive results from the successful projects studied.&lt;ref&gt;See also Raimondo (2016).&lt;/ref&gt; To what extent might citizen-directed subsidies for local media as suggested here improve the demand for (and the supply of) better evaluations, leading to better programs and the lower crime, etc., that came from those programs?  To what extent might this effect be quantified using randomized controlled trials comparing different jurisdictions, analogous to the research for which Banerjee, Duflo, and Kramer won the 2019 Nobel Memorial Prize in Economics?  

This discussion makes us wonder if better research and better news might deliver dramatically more benefits than costs in reducing money wasted on both funding wasteful programs and on failing to fund effective ones?  In particular, might society benefit from matching the 1 percent of the workforce occupied by accountants and auditors with better research and citizen-directed subsidies for news (see Figure 10)?  If, for example, 1 or 2 percent of GDP distributed to local news nonprofits via local elections, as described above, increased the average rate of economic growth in GDP per capita by 0.1 percentage point per year, that increase would accumulate over time, so that after 10 or 20 years, the news would in effect become free, paid by money that implementing political jurisdictions would not have without those subsidies. Moreover those accumulations might remain as long as they were not wiped out by events comparable to the economic disasters documented above in discussing &quot;Stalin and Putin&quot; -- and maybe not even then as suggested by Figure 7. 

=== Other recommendations and natural experiments === 

Table 1 compares the recommendations of McChesney and Nichols (2021, 2022) and Rolnik et al. (2019) with other possible points of reference. Crudely similar to McChesney, Nichols, and Rolnik et al., Karr (2019) and Karr and Aaron (2019) recommend &quot;a 2 percent ad tax on all online enterprises that in 2018 earned more than $200 million in annual digital-ad revenues&quot;. They claim that this &quot;would yield more than $1.8 billion a year&quot;,  which is very roughly 0.008 percent of GDP, $5 per person per year;&lt;ref name=Karr&gt;Karr (2019), Karr and Aaron (2019). US GDP for 2019 was $21,381 billion, per International Monetary Fund (2023). Thus, $1.8 billion is 0.0084% of US GDP and $5.44 for each of the 330,513,000 humans in the US in 2019; round to 0.008% and $5 per capita.&lt;/ref&gt; Google has negotiated agreements similar to this with the governments of Australia and Canada.&lt;ref&gt;Hermida (2023).&lt;/ref&gt; Other points of reference include the percent of GDP devoted to accounting and auditing and advertising. As displayed in Figure 10, accountants and auditors are roughly 1 percent of the workforce in the US. It's not clear how to translate that into a percent of GDP, but 2 percent seems like a reasonable approximation, if we assume that average income of accountants and auditors is a little above the national norm and overhead is not quite double their salaries; this may be conservative, because many accountants and auditors have support staff, who are not accountants nor auditors but support their work. 

Another point of reference is the average annual growth rate in GDP per capita since World War II: A subsidy of 2 percent of GDP would be roughly one year's increase in average annual income since World War II, as noted with Figure 1 above. More precisely, the US economy (GDP per capita adjusted for inflation) was 2.2 percent per year between between 1950 and 1990 but only 1.1 percent between 1990 and 2022, according to inequality expert [[w:Thomas Piketty|Thomas Piketty]], who attributed that slowing in the rate of economic growth to the increase in income inequality in the US since 1975, documented in Fgure 6 above. Whether Piketty is correct or not, if 2 percent per year subsidies for journalism close the gap between 1.1 and 2.2 percent per year, those media subsidies would effectively become free after two years, paid out of income the US would not have without them. This reinforces the main point of this essay regarding the need for randomized controlled trials on any intervention with a credible claim to improving the prospects for broadly shared economic growth for the long term.&lt;ref name=GDP&gt;The growth in US GDP per capita is discussed in the working paper on Wikiversity titled, &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;, accessed 2021-05-19.   For a similar comment about an intervention that increased the rate of economic growth becoming free, paid out of income we would not have without it, has been made about the impact of improving education by {{cite Q|Q56849246}}&lt;!-- Endangering Prosperity: A Global View of the American School--&gt;, p. 12.&lt;/ref&gt;    
This table includes other interventions for which humanity would benefit from more substantive evaluation of their impact. This includes [[w:Democracy voucher|Seattle's &quot;Democracy Voucher&quot; program]], which gives each registered voter four $25 vouchers, totaling $100, which they could give to eligible candidates running for municipal office. However, only the first 47,000 were honored;  this limited the city's commitment to $4.7 million every other year.&lt;ref name=Berman&gt;Berman (2015). The Wikipedia article on [[w:Seattle]] says that the gross metropolitan product (GMP) for the Seattle-Tacoma metropolitan area was $231 billion in 2010 for a population of 3,979,845.  That makes the GMP per capita roughly $58,000.  However, the population of Seattle proper was only 608,660 in 2010, making the Gross City Product roughly $35 billion.  $4.7 million is 0.0133 percent of $35 billion.  However, that's very other year, so it's really only 0.007 percent of the Gross City Product.&lt;/ref&gt; If Seattle can afford $100 per registered voter, many other governmental entities can afford something very roughly comparable for each adult in their jurisdiction. Seattle's &quot;democracy vouchers&quot; are used to fund political campaigns, not local media; they are mentioned here as a point of comparison. 

Other interventions that seem to deserve more research than we've seen are the [[w:New Jersey Civic Information Consortium|New Jersey Civic Information Consortium]] (NJCIC)&lt;ref name=njcicBudget&gt;Karr (2020). Per [[w:New Jersey]] the Gross State Product in 2018 was roughly $640 billion;  it's population in 2020 was roughly 9.3 million. The initial  $500,000 for the project is only $0.05 per person per year and only 0.00008 percent of $640 billion.&lt;/ref&gt; and a program in California to improve local news in communities in dire need of strong local journalism. The NJCIC was initially funded at $500,000, which is only 0.00008 percent of New Jersey economy (GDP) of $640 billion. In 2022, the state of California authorized $25 million for up to 40 Berkeley local news fellowships offering &quot;a $50,000 annual stipend [for 3 years] to supplement their salaries while they work in California newsrooms covering communities in dire need of strong local journalism.&quot; This Berkeley program is roughly $0.21 per person per year, 0.0007 percent of the Gross State Produce of $3.6 trillion that year, for an annual rate of very roughly 0.0002 percent of the Gross State Product.&lt;ref name=Berkeley&gt;Natividad (2023) discusses the Berkeley local news fellowships. California Gross State Product from US Bureau of Economic Analysis (2023). California population on 2022-07-01 from US Census Bureau (2023).&lt;/ref&gt; A similar project in Indiana funded by philanthropies began as the Indiana Local News Initiative&lt;ref&gt;Greenwell (2023).&lt;/ref&gt; and has morphed into Free Press Indiana.&lt;ref&gt;See &quot;[https://www.localnewsforindiana.org LocalNewsForIndiana.org]&quot;; accessed 29 December 2023.&lt;/ref&gt;

Some local [[w:League of Women Voters|Leagues of Women Voters]] have all-volunteer teams who observe official meetings of local governmental bodies and write reports.&lt;ref&gt;Wilson (2007).&lt;/ref&gt; The [[w:City Bureau|City Bureau]] nonprofit news organization in Chicago, Illinous, &quot;trains and pays community members to attend local government meetings and report back on them.&quot;&lt;ref&gt;See &quot;[https://www.citybureau.org/documenters-about citybureau.org/documenters-about]&quot;.&lt;/ref&gt; The program has been so successful, it has exanded to other cities.&lt;ref&gt;Greenwell (2023).&lt;/ref&gt;

For an international comparison, we include [[w:amaBhungane|amaBhungane]],&lt;ref name=amaBhu&gt;The budget for [[w:AmaBhungane#Budget|amaBhungane]] in 2020 was estimated at 590,000 US dollars at the current exchange rate, per analysis in the [[w:AmaBhungane#Budget|budget]] section of the Wikipedia article on amaBhungane. That's 0.00017 percent of South African's nominal GDP for that year of 337.5 million US dollars, per the section on &quot;[[w:Economy of South Africa#Historical statistics 1980–2022|Historical statistics 1980–2022]]&quot; in the Wikipedia article on [[w:Economy of South Africa|Economy of South Africa]]; round that to 0.002 percent for convenience. The population of South Africa that year was estimated at 59,309,000, according to the section on &quot;[[w:Demographics of South Africa#UN Age and population estimates: 1950 to 2030|UN Age and population estimates: 1950 to 2030]]&quot; in the Wikipedia article on [[w:Demographics of South Africa|Demographics of South Africa]]; this gives a budget of 1 penny US per capita. (All these Wikipedia articles were accessed 2023-12-28.)&lt;/ref&gt; whose investigative journalism exposed a corruption scandal that helped force South African President [[w:Jacob Zuma|Jacob Zuma]] to resign in 2018; amaBhungane's budget is very roughly one penny US per person per year in South Africa, 0.0002 percent of GDP. To the extent that this essay provides a fair and balanced account of the impact of journalism on political economy, South African and the rest of the world would benefit from more funding for amaBhungane and other comparable investigative journalism organizations. This could initially include randomized controlled trials involving citizen-directed subsidies for local news outlets in poor communities in South Africa and elsewhere, as we discuss further in the rest of this essay. Without such experiments, we are asking for funds based more on faith than science. 

{| class=&quot;wikitable sortable&quot; 
!option / reference
!% of GDP 
!colspan=2|US$
!per …
!
|-
|style=&quot;text-align:left;&quot;|US postal subsidies for newspapers 1840-44 
| 0.21%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $140
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
| &lt;ref name=McC-N2010/&gt;
|-
|style=&quot;text-align:left;&quot;|McChesney &amp; Nichols (2021, 2022)
| 0.15%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $100
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
| &lt;ref name=McC-N2021/&gt;
|-
|[[Confirmation bias and conflict#Relevant research|Rolnik et al.]] 
| 0.08%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $50
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|adult &amp; year
| &lt;ref name=Rolnik/&gt; 
|-
|[[w:Free Press (organization)|Free Press]]
| 0.008%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $6
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref name=Karr/&gt;
|-
|[[w:Democracy vouchers|Democracy vouchers]]
| 0.007% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $100
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|voter &amp; municipal election for the first 47,000
|&lt;ref name=Berman/&gt;
|-
|[[Confirmation bias and conflict#Advertising and accounting|advertising]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref name=ads/&gt;
|-
|[[Confirmation bias and conflict#Advertising and accounting|accounting]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref&gt;As noted with Figure 10 and the discussion above, accountants and auditors are roughly 1 percent of the US workforce, and it seems reasonable to guess that their pay combined with support staff and overhead would likely make them roughly double that, 2 percent, as a portion of GDP.&lt;/ref&gt;
|-
|[[US Gross Domestic Product (GDP) per capita|US productivity improvements]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year (GDP per capita)
|&lt;ref&gt;For an analysis of the rate of growth in US GDP per capita, see the working paper on Wikiversity titled, &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;&lt;/ref&gt;
|-
|[[w:New Jersey Civic Information Consortium|New Jersey Civic Information Consortium]]
| 0.00008%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .05
|person &amp; year
|&lt;ref name=njcicBudget/&gt;
|-
| Berkeley local news fellowships  
| 0.0002%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .21
|person &amp; year
|&lt;ref name=Berkeley/&gt;
|-
|[[w:amaBhungane|amaBhungane]]
| 0.0002%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .01
|person &amp; year in South Africa
|&lt;ref name=amaBhu/&gt;
|}
Table 1. Media subsidies and other points of reference. At the low end, political corruption exposed in part by amaBhungane forced the resignation in 2018 of South African President Zuma on a budget that's very roughly one penny US per person per year. If much higher subsidies of 1 percent of GDP restored an annual growth rate of 2.2 percent per year from the more recent 1.1 percent discussed with Figure 1 above, those subsidies would pay for themselves from one year's growth that the US would not otherwise have.

== Other factors == 

We feel a need here to suggest other issues to consider in designing experiments to improve the political economy:  education, empowering women, free speech, free press, peaceful assembly, and reducing political polarization.

EDUCATION:  Modern research suggests that society might have lower crime&lt;ref&gt;Wang et al. (2022).&lt;/ref&gt; and faster rates of economic growth with better funding for and better research&lt;ref&gt;Hanushek and Woessmann (2015).&lt;/ref&gt; on quality child care from pregnancy through age 17. 

EMPOWERING WOMEN:  Might the best known way to limit and reverse population growth be to empower women and girls?  Without that, might the human population continue to grow until some major disaster reduces that population dramatically?&lt;ref&gt;Roser (2017).&lt;/ref&gt;

FREE SPEECH, FREE PRESS, PEACEFUL ASSEMBLY:  Verbitsky said, &quot;Journalism is disseminating information that someone does not want known; the rest is propaganda.&quot;&lt;ref&gt;Verbitsky (2006, p. 16), author's translation from Spanish.&lt;/ref&gt; This discussion of threats, arrests, kidnappings, and murders of journalists&lt;ref&gt;Monitored especially by the [[w:Committee to Protect Journalists|Committee to Protect Journalists]], as discussed in the Wikipedia article on them, accessed 2023-07-04.&lt;/ref&gt; and violent suppression of peaceful assemblies&lt;ref&gt;Monitored by Freedom House and others.  See, e.g., the Wikipedia article on &quot;[[w:List of freedom indices|List of freedom indices]]&quot;, accessed 2023-07-04.&lt;/ref&gt; encourages us to consider the potential utility of efforts to improve local news, as noted by contributors to Islam et al., eds. (2002), cited above.  Data on such problems should be considered in selecting sites for experiments with citizen-directed subsidies for journalism and in analyzing the results from such experiments. Such data should include the incidence of legal proceedings against journalists and publishers&lt;ref&gt;Including the risks of [[w:trategic lawsuit against public participation|strategic lawsuits against public participation]] (SLAPPs) and other questionable uses of the courts including some documented in the &quot;[[w:Freedom of the Press Foundation#U.S. Press Freedom Tracker|U.S. Press Freedom Tracker]]&quot;, mentioned above.&lt;/ref&gt; as well as threats, murders, etc., in jurisdictions comparable to experimental jurisdictions.  

REDUCING POLITICAL POLARIZATION:  What interventions might be tested that would attempt to reduce political polarization while also experimenting with increasing funding for news through small, diverse news organizations? For example, might an ad campaign feature someone saying, &quot;We don't talk politics&quot;, with a  reply, &quot;We have to talk politics with humility and mutual respect, because the alternative is killing people over misunderstandings&quot;? Might another ad say, &quot;Don't get angry: Get curious&quot;?  What can be done to encourage people to get curious rather than angry when they hear something that contradicts their preconceptions?  How can people be encouraged to talk politics with humility and respect for others, understanding that everyone can be misinformed and others might have useful information?&lt;ref&gt;Wikiversity &quot;[[How can we know?]]&quot;, accessed 2023-07-22, reviews relevant research relating to political polarization.  Yom-Tov et al. (2018) described a randomized-controlled trial that compared the effectiveness of different advertisements &quot;to improve food choices and integrate exercise into daily activities of internet users.&quot; They found &quot;powerful ways to measure and improve the effectiveness of online public health interventions&quot; and showed &quot;that corporations that use these sophisticated tools to promote unhealthy products can potentially be outbid and outmaneuvered.&quot; Similar research might attempt to promote strategies for countering political polarization.  See also Piwowarski  et al. (2019).&lt;/ref&gt;  

FOCUS ON POLITICIANS: Mansuri et al. (2023) randomly assigned presidents of village governments in the state of [[w:Tamil Nadu|Tamil Nadu]] in India to one of three groups with (1) a financial incentive or (2) a certificate with an information campaign (without a financial incentive) for better government or (3) a control group. They found that the public benefitted from both the financial and non-financial incentives, and the non-financial incentives were more cost effective. Might it make sense in some experimental jurisdictions to structure the subsidies for local news by asking voters to allocate, e.g., half their votes for local news to outlet(s) that they think provide the best information about politicians with the other half based on &quot;general news&quot;?&lt;ref&gt;Mansuri et al. (2023).&lt;/ref&gt;

PIGGYBACK ON COMMUNITY AND LOCAL DEVELOPMENT PROGRAMS:  The World Bank (2023) notes that, &quot;Experience has shown that when given clear and transparent rules, access to information, and appropriate technical and financial support, communities can effectively organize to identify community priorities and address local development challenges by working in partnership with local governments and other institutions to build small-scale infrastructure, deliver basic services and enhance livelihoods. The World Bank recognizes that CLD [Community and Local Development]  approaches and actions are important elements of an effective poverty-reduction and sustainable development strategy.&quot; This suggests that experiments in citizen-directed subsidies for news might best be implemented as adjuncts to other CLD projects to improve &quot;access to information&quot; needed for the success of that and follow-on projects.  Such news subsidies should complement and reinforce the quality of monitoring and evaluation, which was &quot;significantly and positively associated with project outcome as institutionally measured at the World Bank&quot;.&lt;ref&gt;Raimondo (2016). See also Bedasso (2021).&lt;/ref&gt;

Simple experiments of the type suggested here might add citizen-directed subsidies for local news to a random selection of Community and Local Development (CLD) projects.  Such experimental jurisdictions should be small enough that the budget for the proposed citizen-directed subsidies for news would be seen as feasible but large enough so appropriate data could be obtained and compared with control jurisdictions not receiving such subsidies for local news.

However, some potential recipients of CLD funding may be in news deserts or with &quot;ghost newspapers&quot;, as mentioned above.  Some may not have at least three local news outlets that have been publishing something they call news each workday for at least six months, as required for the local elections recommended by McChesney and Nichols (2021, 2022), outlined above.  In such jurisdictions, the local consultations that identify community priorities for CLD funding should also include discussions of how to grow competitive local news outlets to help the community maximize the benefits they get from the project.  

The need for &quot;at least three local news outlets&quot; is reinforced by the possibilities that two or three local news outlets may be an [[w:oligopoly|oligopoly]], acting like a monopoly.  This risk may be minimized by working to ''limit'' barriers to entry and to encourage different news outlets to serve different segments of the market for news.  The risks of oligopolistic behavior may be further reduced by requiring all recipients of citizen-directed subsidies to release their content under a free license like the Creative Commons Attribution-ShareAlike (SS BY-SA) 4.0 international license.  This could push each independent local news outlet to spend part of their time reading each other's work while pursuing their own journalistic investigations, hoping for scoops that could attract a wider audience after being cited by other outlets.&lt;ref&gt;Wikipedia &quot;[[w:Oligopoly|Oligopoly]]&quot;, accessed 2023-07-06.&lt;/ref&gt;

This preference for at least three independent local news outlets in an experimental jurisdiction puts a lower bound on the size of jurisdictions to be included as experimental units, especially if we assume that the independent outlets should employ on average at least two journalists, giving a minimum of six journalists employed by local news outlets in an experimental jurisdiction.  The discussions above suggested subsidies ranging from 0.08 percent to 2 percent or more.  To get a lower bound for the size of experimental jurisdictions, we divide 6 by 0.08 and 2 percent:  Six journalists would be 0.08 percent of a population of 7,500 and 2 percent of a population of 300.

== Sampling units / experimental polities == 

Many local governments could fund local news nonprofits at 0.15% of GDP, because it would likely be comparable to what they currently spend on accounting, media and public relations.&lt;ref&gt;&quot;State and local governments [in the US] spent $3.5 trillion on direct general government expenditures in fiscal year 2020&quot;, with states spending $1.7 trillion and local governments $1.8 trillion&quot;, per Urban Institute (2024). The nominal GDP of the US for 2020 was $21 trillion, per International Monetary Fund (2024). Thus, local government spending $1.7 trillion is 1.8% of the $21 trillion US GDP, which is comparable to the money spent on accounting per Figure 10 and advertising per Figure 9.&lt;/ref&gt; If the results of such funding are even a modest percent of the benefits claimed in the documents cited above, any jurisdiction that does that would likely obtain a handsome return on that investment. 

Jurisdictions for randomized controlled trials might include some of the members of the United Nations with the smallest Gross Domestic Products (GDPs) or even some of the poorest census-designated places&lt;ref&gt;Wikipedia &quot;[[w:Census-designated place|Census-designated place]]&quot;, accessed 2023-07-11.&lt;/ref&gt; in a country like the US. Alternatively, they might include areas with seemingly intractable cycles of violence like Israel and Palestine: The budget for interventions like those proposed herein are a fraction of what is being spent on defense and on violence challenging existing power structures. If interventions roughly comparable to those discussed herein can reduce the lethality of a conflict at a modest cost, it would have an incredible return on investment (ROI). That would be true not merely for the focus of the intervention but for other similar conflicts. 

For illustration purposes only, Table 2 lists the six countries in the United Nations with the smallest GDPs in 2021 in US dollars at current prices according to the United Nations Statistics Division plus Palestine and Israel, along with their populations and GDP per capita plus the money required to fund citizen-directed subsidies at 0.15 percent of GDP, as recommended by McChesney and Nichols (2021, 2022).  The rough budgets suggested here would only be for a news subsidy companion to Community and Local Development (CLD) projects, as discussed above or for intervention(s) attempting to reduce the lethality of conflict.  Other factors should be considered in detailed planning.  For example, the budget for such a project in [[w:Montserrat|Montserrat]] may need to be increased to support greater diversity in the local news outlets actually subsidized.  And a careful study of local culture in [[w:Kiribati|Kiribati]] may indicate that the suggested budget figure there may support substantially fewer than the 97 journalists suggested by the naive computations in this table.  The key point, however, is that subsidies of this magnitude would be modest as a proportion of many other projects funded by agencies like the World Bank or the money spent on defense or war. 

{| class=&quot;wikitable sortable&quot; style=text-align:right
! Country !! Population !! GDP / capita
! GDP (million USD) 
! annual subsidy at 0.15% of GDP ($K) 
! number of journalists&lt;sup&gt;(*)&lt;/sup&gt;
|-
| [[w:Tuvalu|Tuvalu]] || 11,204 || $5,370 || $60 || $90 ||8.4
|-
| [[w:Montserrat|Montserrat]] || 4,417 || $16,199 || $72 || $107 || 3.3
|- 
| [[w:Nauru|Nauru]] || 12,511 || $12,390 || $155 || $233 || 9.4
|- 
| [[w:Palau|Palau]] || 18,024 || $12,084 || $218 || $327 || 13.5
|- 
| [[w:Kiribati|Kiribati]] || 128,874 || $1,765 || $227 || $341 ||96.7
|- 
| [[w:Marshall Islands|Marshall Islands]] || 42,040 || $6,111 || $257 || $385 || 31.5
|- 
| [[w:State of Palestine|State of Palestine]] || 5,483,450 || $3,302 || $18,037 || $27,055 || 4,113
|- 
| [[w:Israel| Israel]] || 9,877,280 || $48,757 || $481,591 || $722,387 || 7,408 
|}
Table 2.  Rough estimate of the budget for subsidies at 0.15 percent of GDP for the 6 smallest members of the UN plus Palestine and Israel. Population and GDP at current prices per United Nations Statistics Division (2023). (*) &quot;Number of journalists&quot; was computed assuming each journalist would cost twice the GDP / capita.  For example, the GDP / capita for Tuvalu in this table is $5,370. Double that to get $10,740.  Divide that into $90,000 to get 8.4.  

Other possibilities for experimental units might be historically impoverished subnational groups like [[w:Native Americans in the United States|Native American jurisdictions in the United States]].  As of 12 January 2023 there were &quot;574 Tribal entities recognized by and eligible for funding and services from the [[w:Bureau of Indian Affairs|Bureau of Indian Affairs]] (BIA)&quot;, some of which have multiple subunits, e.g., populations in different counties or census-designated places. For example, the largest is the [[w:Navajo Nation|Navajo Nation Reservation]] that is split between Arizona, New Mexico, and Utah.&lt;ref&gt;Newland (2023).&lt;/ref&gt; Some of these subdivisions are too small to be suitable for experiments in citizen-directed subsidies for news.  Others have subdivisions large enough so that some subdivisions might be in experimental group(s) with others as controls.  If there are at least three subdivisions with sufficient populations, at least one could be a control, with others being Community and Local Development (CLD) projects both with and without companion news subsidies, as discussed above.&lt;ref&gt;Data analysis might consider spatial autocorrelation, as used by Mohammadi et al. (2022) and multi-level time series text analysis, used by Friedland et al. (2022).  The latter discuss &quot;Asymmetric communication ecologies and the erosion of civil society in Wisconsin&quot;:  That state had historically been moderate &quot;with a strong progressive legacy&quot;.  Then in 2010 they elected a governor who attacked the state's public sector unions with substantial success and voted for Donald Trump for President in 2016 but ''against'' him in 2020.&lt;/ref&gt;

== Supplement not replace other funding == 

The subsidies proposed here should supplement not replace other funding, similar to the subsidies under the US Postal Service Act of 1792.  McChesney and Nichols recommended that an organization should be publishing something they call news five days per week for at least six months, so the voters would know what they are voting for.  Those criteria might be modified, at least in some experimental jurisdictions, especially in news deserts, as something else is done to create local news organizations eligible to receive a portion of the experimental citizen-directed subsidies.  

The [[w:Institute for Nonprofit News|Institute for Nonprofit News]] and Local Independent Online News (LION) Publishers&lt;ref&gt;&lt;!-- Local Independent Online News (LION) Publishers--&gt;{{cite Q|Q104172660}}&lt;/ref&gt; help local news organizations get started and maintain themselves.  Organizations like them might help new local news initiatives in experimental jurisdictions as discussed in this article.

== Funding research in the value of local news ==

We would expect two sources of funding for research to quantify the value of local news 

* The [[w:World Bank|World Bank]] has already discussed the value of news. We would expect that organizations that fund community and local development projects would also want to fund experiments in anything that seemed likely to increase the return on their investments in such projects. 
* Folkenflik (2023) wrote, &quot;Some of the biggest names in American philanthropy have joined forces to spend at least $500 million over five years to revitalize the coverage of local news in places where it has waned.&quot; This group of philanthropic organizations includes the American Journalism Project, which says they &quot;measure the impact of our philanthropic investments and venture support by evaluating our efficacy in catalyzing grantees’ organizational growth, sustainability and impact.&quot;&lt;ref&gt;Website of the American Journalism Project accessed 29 December 2023 ([https://www.theajp.org/about/impact/# https://www.theajp.org/about/impact/#]).&lt;/ref&gt; 

If the claims made above for the value of news are fair, then appropriate experiments might be able to quantify who benefit from improving the news ecology, how much they benefit, which structures seem to work the best, and even the optimal level of funding.

== Summary == 

This article has summarized numerous claims regarding different ways in which information is a public good.  Many such claims can be tested in experiments crudely similar to those for which Banerjee, Duflo, and Kremer won the [[w:2019 Nobel Memorial Prize in Economic Sciences|2019 Nobel Memorial Prize in Economics]].  We suggest funding such projects as companions to Community and Local Development (CLD) projects.  If the research cited above is replicable, the returns on such investments could be huge, delivering benefits to the end of human civilization, similar to those claimed for newspapers published in the US in the early nineteenth century, which may have made major contributions to carrying the US to its current position of world leadership and to developing technologies that benefit the vast majority of humanity the world over today.

== Acknowledgements == 

Thanks especially to Bruce Preville who pushed for evidence supporting wide ranging claims of media influence in limiting progress against many societal ills.  Thanks also to Dave Black for suggesting experimenting with Native American jurisdictions in the US and to Joy Ellsworth for describing the substantial cultural challenges that such interventions might face.

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* &lt;!-- Brayton Harris (1999) Blue &amp; gray in black &amp; white: newspapers in the Civil War (Brassey's) --&gt;{{cite Q|Q114731128}}
* &lt;!--Alfred Hermida (2023) Google’s $100 million to Canada’s news industry is a small price to pay to avoid regulation, The Conversation--&gt;{{cite Q|Q124052109}}
* &lt;!-- Jim Hightower (2023)  Breaking News: New Life for Local Newspapers--&gt;{{cite Q|Q121438032}}
* &lt;!-- Victoria Holmes (15 November 2022). &quot;The Kansas City Defender, reaching young Black communities currently underserved by mainstream media&quot;. Editor &amp; Publisher --&gt;{{cite Q|Q116194691}}
* &lt;!-- International Monetary Fund (2023) &quot;World Economic Outlook database: April 2023&quot; --&gt;{{cite Q|Q120755124|author=International Monetary Fund|publisher=unset|date=2023}}, accessed 2023-04.
* &lt;!--  Roumeen Islam, Simeon Djankov, and Caralee McLiesh, eds. (2002) The Right to Tell: The Role of Mass Media in Economic Development (World Bank) --&gt;{{cite Q|Q118935200}}
* &lt;!--Junaid Jahangir (2023)  A Note on Teaching Economic Inequality--&gt;{{cite Q|Q124004544}}
* &lt;!-- Dominic D. P. Johnson (2004). Overconfidence and War: The Havoc and Glory of Positive Illusions (Harvard U. Pr.) --&gt;{{cite Q|Q118106389}}
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*  &lt;!-- Karr (2020)New Jersey Funds the Civic Info Consortium, Recognizes the Vital Role Local News Plays During Crises --&gt;{{cite Q|Q106146745}}
* &lt;!--Karr and Aaron (2019) Beyond Fixing Facebook --&gt;{{cite Q|Q104624308}}
* &lt;!-- Mahsa Kaviani, Lily Yuanzhi Li, and Hosein Maleki (2022) Media, Partisan Ideology, and Corporate Social Responsibility--&gt;{{cite Q|Q121493540}}
* &lt;!--  John Kennedy (1962) Address on the First Anniversary of the Alliance for Progress--&gt;{{cite Q|Q19030219}}
* &lt;!-- Mary Elizabeth King (2007) A quiet revolution: the first Palestinian Intifada and nonviolent resistance--&gt;{{cite Q|Q123299945}}
* &lt;!-- Phillip Knightley (2004) The first casualty: the war correspondent as hero and myth-maker from the Crimea to Iraq (Johns Hopkins U. Pr.) --&gt;{{cite Q|Q118227485}}
* &lt;!-- Greg Leiserson and Danny Yagan (2021-09-23) &quot;What Is the Average Federal Individual Income Tax Rate on the Wealthiest Americans?&quot;, The White House --&gt;{{cite Q|Q118192958}}
* &lt;!-- Erik Lindahl (1919) Die gerechtigkeit der besteuerung--&gt;{{cite Q|Q124720010}}
* &lt;!-- Erik Lindahl (1958) Just Taxation: A Positive Solution--&gt;{{cite Q|Q124720029}}
* &lt;!-- Mark Lloyd and Lewis A. Friedland, eds. (2016) The Communication Crisis in America, And How to Fix It (Palgrave Macmillan) --&gt;{{cite Q|Q56081584|authors=Mark Lloyd and Lewis A. Friedland, eds.}}
* &lt;!-- Elizabeth Loftus and Willem Albert Wagenaar (1988) &quot;Lawyer's predictions of success&quot;--&gt;{{cite Q|Q120793060}}
* &lt;!-- Los Angeles Times (2023-03-27) &quot;The California newspaper that has no reporters left&quot;--&gt;{{cite Q|Q120766479|author=''Los Angeles Times''|journal=unset}}
* &lt;!-- Lisa Loving (2019) Street Journalist--&gt;{{cite Q|Q65115459}}
* &lt;!--Ghazala Mansuri, Nethra Palaniswamy, Vijayendra Rao, Slesh A. Shrestha (2023-08) &quot;Money versus Kudos: The impact of incentivizing local politicians in India&quot;, Journal of Public Economics, Volume 224, August 2023, 104922--&gt;{{cite Q|Q124058948}}
* &lt;!-- Robert W. McChesney (2004) The Problem of the Media: U.S. Communication Politics in the 21st Century (Monthly Review Pr.) --&gt;{{cite Q|Q7758439}}
* &lt;!-- Robert W. McChesney (2015) Rich Media, Poor Democracy: Communications politics in dubious times, 3rd ed.--&gt;{{cite Q|Q118219867}}
* &lt;!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) --&gt;{{cite Q|Q104888067}}.
* &lt;!-- Robert W. McChesney; John Nichols (2016). People get ready--&gt;{{cite Q|Q87619174}}
* &lt;!-- Robert W. McChesney; John Nichols (2021). &quot;The Local Journalism Initiative: a proposal to protect and extend democracy&quot;. Columbia Journalism Review, 30 November 2021 --&gt;{{cite Q|Q109978060}}
* &lt;!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) --&gt;{{cite Q|Q109978337}}
* &lt;!-- H. R. McMaster (2020) Battlegrounds: The Fight to Defend the Free World (HarperCollins) --&gt;{{cite Q|Q104774898}}
* &lt;!-- Antonela Miho (2020) Small Screen, Big Echo? Estimating the Political Persuasion of Local Television News Bias using Sinclair Broadcast Group as a Natural Experiment--&gt;{{cite Q|Q120895467}}
* &lt;!-- Hosein Mohammadi, Samira Shayanmehr and Juan D. Borrero (2023) &quot;Does Freedom Matter for Sustainable Economic Development? New Evidence from Spatial Econometric Analysis&quot;, Mathematics, 11 --&gt;{{cite Q|Q120736319}}
* &lt;!-- US National Academies of Sciences, Engineering, and Medicine (2023) Nobel Prize Summit Fuels Initiatives to Combat Misinformation and Disinformation and Build Trust in Science--&gt;{{cite Q|Q124711722|author=National Academies}}
* &lt;!-- Ivan Natividad (2022) State funds Berkeley Journalism $25 million to strengthen California's local news coverage--&gt;{{cite Q|Q123537590}}
* &lt;!-- Bryan Newland (2023) &quot;Indian Entities Recognized by and Eligible To Receive Services From the United States Bureau of Indian Affairs&quot;, Bureau of Indian Affairs, US Department of the Interior --&gt;{{cite Q|Q120512083}}
* &lt;!-- Jeremy Nguyen, Abbas Valadkhani, Alan Nguyen, Alexandra Wake (2021) Press Freedom and the Global Economy: The Cost of Slipping Backwards--&gt;{{cite Q|Q121011930}}
* &lt;!-- The Nobel Prize (2019) &quot;Prize in economic sciences 2019&quot; --&gt;{{cite Q|Q119948228|author=Nobel Prize}}
* &lt;!--  Ned Parker, Isabel Coles, Raheem Salma (2014) &quot;Special Report: How Mosul fell - An Iraqi general disputes Baghdad's&quot;, Reuters--&gt;{{cite Q|Q124326278}}
* &lt;!-- M. Piwowarski, U. Shankar Singh, &amp; K. Nermend (2019). &quot;The cognitive neuroscience methods in the analysis of the impact of advertisements in shaping people's health habits&quot; European Research Studies Journal, 22(4), 457-471 --&gt;{{cite Q|Q118087826}}
* &lt;!-- Thomas Piketty (2022) A brief history of equality (Harvard U. Pr.) --&gt;{{cite Q|Q115434513}}
* &lt;!-- Scott Plous (1993) The psychology of judgment and decision making (McGraw-Hill) --&gt;{{cite Q|Q118105219}}
* &lt;!--Kyle Pope (2023-04-17) &quot;Free Press, Functioning Democracy: You can’t have one without the other&quot;, Columbia Journalism Review--&gt;{{cite Q|Q124047437}}
* &lt;!-- Gary W. Potter and Victor E. Kappeler (1998) Constructing Crime: Perspectives on Making News and Social Problems (Waveland Pr.) --&gt;{{cite Q|Q96343487|author=Gary W. Potter and Victor E. Kapeller, eds.}}
* &lt;!-- Estelle Raimondo (2016) &quot;What Difference Does Good Monitoring and Evaluation Make to World Bank Project Performance?&quot;, Policy Research Working Paper 7726, World Bank --&gt;{{cite Q|Q120750870}}
* &lt;!-- Reuters (2023-10-25) Israel to amend budget, Gaza war direct cost at $246 mln daily--&gt;{{cite Q|Q124336510|author=Reuters}}
* &lt;!-- Guy Rolnik; Julia Cagé; Joshua Gans; Ellen P. Goodman; Brian G. Knight; Andrea Prat; Anya Schiffrin (1 July 2019), &quot;Protecting Journalism in the Age of Digital Platforms&quot; (PDF), Booth School of Business --&gt;{{cite Q|Q106465358}}
* &lt;!-- Max Roser (2017) &quot;Fertility Rate&quot;, Our World in Data --&gt;{{cite Q|Q120333634}}
* &lt;!-- Vincent F. Sacco (1995) &quot;Media Constructions of Crime&quot;, Annals Of The American Academy of Political and Social Science, 539:141-154, Wikidata Q56805896, reprinted as ch. 2 of Potter and Kappeler (1998) --&gt;{{cite Q|Q106878177}}, reprinted as ch. 2 in Potter and Capeller (1998).
* &lt;!-- Vincent F. Sacco (2005) When Crime Waves (SAGE) --&gt;{{cite Q|Q96344789}}
* &lt;!-- Fabian Scheidler (2024-03a) Espoirs et misère de la critique des médias en Allemagne, Le Monde diplomatique--&gt;{{cite Q|Q125430621|date=2024a}}
* &lt;!-- Fabian Scheidler (2024-03b) &quot;La disparition d'une école contestataire&quot;, Le Monde diplomatique--&gt;{{cite Q|Q125455715|date=2024b}}
* &lt;!-- Sam Schulhofer-Wohl and Miguel Garrido (2009) &quot;Do newspapers matter?  Short-run and long-run evidence from the closure of the Cincinnati Post&quot;, NBER Working Paper Series 14817 (http://www.nber.org/papers/w14817, 2023-05-05) --&gt;{{cite Q|Q105879512}}
* &lt;!-- Lawrence J. Schweinhart, Jeanne Montie, Zongping Xiang, W. Steven Barnett, Clive R. Belfield, and Milagros Nores (2005) &quot;Lifetime effects: The High/Scope Perry Preschool Study through age 40: Summary, Conclusions, and Frequently Asked Questions&quot;, HighScope --&gt;{{cite Q|Q119527802}}
* &lt;!--Avi Shlaim (2014) The Iron Wall: Israel and the Arab World, 2nd ed. (Penguin)--&gt;{{cite Q|Q124213243}}
* &lt;!-- Joseph Sparling and Kimberly Meunier (24 July 2019). &quot;Abecedarian: An Early Childhood Education Approach that has a Rich History and a Vibrant Present&quot;. International Journal of Early Childhood. 51: 207–216. ISSN 0020-7187--&gt;{{cite Q|Q120552586}}
* &lt;!-- Josh Stearns and Christine Schmidt (2022) How We Know Journalism is Good for Democracy--&gt;{{cite Q|Q123522369}}
* &lt;!-- Joseph E. Stiglitz (1999) &quot;Knowledge as a global public good&quot;, chapter in Kaul et al. (1999), pp. 308-315 --&gt;{{cite Q|Q120732906}}
* &lt;!-- Joseph Stiglitz (2002) Transparency in Government--&gt;{{cite Q|Q120941389}}
* &lt;!-- Leigh Thompson (2020) Heart and mind of the negotiator, 7th ed. --&gt;{{cite Q|Q120870057}}
* &lt;!-- Leigh Thompson and George Loewenstein (1992) &quot;Egocentric interpretations of fairness and interpersonal conflict&quot;, Organizational Behavior and Human Decision Processes. 51 (2): 176–197--&gt;{{cite Q|Q29300051}}
* &lt;!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) --&gt;{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}}
* &lt;!-- Rachel Treisman (2021)  When this hedge fund buys local newspapers, democracy suffers--&gt;{{cite Q|Q121438095}}
* &lt;!-- United Nations (2022) World Population Prospects 2022, Online Edition (Department of Economic and Social Affairs, Population Division) --&gt;{{cite Q|Q113015862|author=United Nations|publisher=unset}}
* &lt;!-- United Nations Statistics Division (2021) &quot;Basic Data Selection&quot; --&gt;{{cite Q|Q120485981|author=United Nations Statistics Division|date=2023-07-06|publisher=unset}}
* &lt;!-- US Bureau of Economic Analysis (2023-03-31) Gross Domestic Product by State, 4th Quarter and Annual 2022--&gt;{{cite Q|Q124036982|author=US Bureau of Economic Analysis}}
* &lt;!-- US Census Bureau (2023) Quick facts--&gt;{{cite Q|Q124037268|author=US Census Bureau}}
* &lt;!--US Institute of Peace (2016) Tools for Improving Media Interventions in Conflict Zones--&gt;{{cite Q|Q124692870|author =US Institute of Peace}}
* &lt;!-- Siva Vaidhyanathan (2018) Antisocial Media: How Facebook Disconnects Us and Undermines Democracy (Oxford U. Pr.) --&gt;{{cite Q|Q56027099}}
* &lt;!-- Alexandra Wake, Abbas Valadkhani, Alan Nguyen, and Jeremy Nguyen (2021-04-15) &quot;The Economic Costs of Curbing Press Freedom&quot;, Global Investigative Journalism Network--&gt;{{cite Q|Q120739929}}
* &lt;!-- Leah Wang, Wendy Sawyer, Tiana Herring and Emily Widra (2022) &quot;Beyond the count: A deep dive into state prison populations&quot;, Prison Policy Initiative --&gt;{{cite Q|Q120332620}}
* &lt;!-- Garrett Watson (2022) Combined Federal and State Corporate Income Tax Rates in 2022 --&gt;{{cite Q|Q120793929}}  
* &lt;!-- Richard G. Wilkinson and Kate E. Pickett (2017) &quot;The Enemy between Us: The Psychological and Social Costs Of Inequality&quot;, European Journal of Social Psychology, 47(1)11-24 --&gt;{{cite Q|Q118385024}}
* &lt;!--Mary G. Wilson (2007) Observing Your Government in Action: Protecting your right to know (LWV)--&gt;{{cite Q|Q124048761}}
* &lt;!-- World Bank (2022) Annual Report --&gt;{{cite Q|Q118270948|author=World Bank|series=unset}}
* &lt;!-- World Bank (2023) &quot;Community and Local Development&quot; (web page:  &quot;https://www.worldbank.org/en/topic/communitydrivendevelopment&quot;) --&gt;{{cite Q|Q120380457|author=World Bank|date=2023-03-27|publisher=unset}}
* &lt;!-- Elad Yom-Tov, Jinia Shembekar, Sarah Barclay, and Peter Muennig (2018) &quot;The effectiveness of public health advertisements to promote health: a randomized-controlled trial on 794,000 participants&quot;--&gt;{{cite Q|Q91869157}}
* &lt;!-- Erica York (2023-01-26) &quot;Summary of the Latest Federal Income Tax Data, 2023 Update&quot;, Tax Foundation --&gt;{{cite Q|Q118189145}}
* &lt;!-- Shoshana Zuboff (2019) The Age of Surveillance Capitalism (PublicAffairs) --&gt;{{cite Q|Q75804726}}
* James D. Wolfensohn (2002) Foreword to Islam et al. (2002, pp. v-vi).

== Notes ==
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[[Category:News]] 
[[Category:Original research]]
[[Category:Research]]
[[Category:Political science]]
[[Category:Media]]
[[Category:Freedom and abundance]]
[[Category:Economics]]
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:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].''  

::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' 

== Abstract == 

This article reviews literature relevant to the claim that &quot;information is a public good&quot; and recommends experiments to quantify the impact of news on society, including on violent conflicts and broadly shared economic growth. We propose randomized controlled trials to evaluate the relative effectiveness of alternative interventions on the lethality of conflict and broadly shared economic growth. Experimental units would be polities in conflict or with incomes (nominal Gross Domestic Products, GDPs or gross local products) small enough so competitive local news outlets could be funded by philanthropies or organizations like the World Bank but large enough that their political economies have been tracked with sufficient accuracy to allow them to be considered in such an experiment. One factor in such experiments would be subsidies for local journalism, perhaps distributed to local news outlets on the basis of local elections, similar to the proposal of McChesney and Nichols (2021, 2022).

== Introduction ==

:''Information is a public good.''&lt;ref&gt;This is the title of Cagé and Huet (2021, in French).  However, the thrust of their book is very different.  It is subtitled, &quot;Refounding media ownership&quot;.  Their focus is on creating legal structure(s) to support journalistic independence as outlined in Cagé (2016).&lt;/ref&gt; 
:''Misinformation is a public nuisance.''&lt;ref&gt;The Wikipedia article on &quot;[[w:Public nuisance|Public nuisance]]&quot; says, &quot;In English criminal law, public nuisance was a common law offence in which the injury, loss, or damage is suffered by the public, in general, rather than an individual, in particular.&quot; (accessed 2023-04-24.)&lt;/ref&gt;
:''Disinformation is a public evil.''&lt;ref&gt;The initial author of this essay is unaware of any previous use of the term, &quot;public evil&quot;, but it seems appropriate in this context to describe content disseminated by mass media, including social media, curated with the explicit intent to convince people to support public policies contrary to the best interests of the audience and the general public.&lt;/ref&gt; 

=== Public goods ===

In economics, a [[w:public good|public good]] is a good (or service) that is both [[w:non-rivalrous|non-rivalrous]] and [[w:non-excludable|non-excludable]].&lt;ref&gt;e.g., Cornes and Sandler (1996).&lt;/ref&gt; Non-rivalrous means that we can all consume it at the same time. An apple is rivalrous, because if I eat an apple, you cannot eat the same apple. 

A printed newspaper may be rivalrous, because it may not be easy for you and me to hold the same sheet of paper and read it at the same time. However, the ''news'' itself is non-rivalrous, because both of us and anyone else can consume the same news at the same time, once it is produced, especially if it's published openly on the Internet or broadcasted on radio or television. 

Non-excludable means that once the good is produced, anyone can use it without paying for it. Information is non-excludable, because everyone can consume it at the same time once it becomes available. [[w:Copyright|Copyright]] law does ''not'' apply to information: It applies to ''expression''.&lt;ref&gt;The US Copyright Act of 1976, Section 102, says, &quot;Copyright protection subsists ... in original works of authorship fixed in any tangible medium of expression ... . In no case does copyright protection ... extend to any idea, procedure, process, system, method of operation, concept, principle, or discovery.&quot; 17 U.S. Code § 102. &lt;!--US Copyright Law of 1976--&gt;{{cite Q|Q3196755}}&lt;/ref&gt; [[w:Joseph Stiglitz|Stiglitz]] (1999) said that [[w:Thomas Jefferson|Thomas Jefferson]] anticipated the modern concept of information as a public good by saying, &quot;He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me.&quot; Stiglitz distinguished between &quot;push and pull mechanisms&quot; to promote innovation and creative work: &quot;Push&quot; mechanisms pay for work upfront, hoping that it will achieve a desired outcome, like citizen-directed subsidies for newspapers. &quot;Pull&quot; mechanisms set a target and then reward those who reach the target, like copyrights and patents.&lt;ref&gt;Baker (2023).&lt;/ref&gt;

Lindahl (1919, 1958) recommended taxing people for public goods in proportion to the benefits they receive. For subsidies for news, especially citizen-directed, this would mean taxing primarily the poor and middle class to fund this.&lt;ref&gt;For more on this, see the Wikipedia articles on [[w:Lindahl tax|Lindahl tax]] and [[w:Theories of taxation|Theories of taxation]].&lt;/ref&gt; If they receive benefits as claimed in the literature cited in this article, the benefits they receive would soon exceed the taxes they pay for it, making the news subsidies effectively free in perpetuity, paid by benefits the poor and middle class would not have without these subsidies. If Piketty (2021, cited below with Figure 1) is correct, the ultra-wealthy would likely also benefit in absolute terms, though the relative distinction between them and the poor would be reduced.

This article recommends [[w:randomized controlled trials|randomized controlled trials]] to quantify the extent to which experimental interventions benefit the public by modifying information environment(s) in ways that (a) reduce political polarization and / or violence and / or (b) improve broadly shared peace and prosperity for the long term.

=== Sharing increases the value === 

The logic behind claiming that &quot;information is a public good&quot; can be easily understood as follows:  

:''If I know the best solution to any major societal problem, it will not help anyone unless a critical mass of some body politic shares that perception.  Conversely, if a critical mass of a body politic believes in the need to implement a certain reform, it will happen, even if I am ignorant of it or completely opposed to it.''  

We can extend this analysis to our worst enemies:  

:It is in ''our best interest'' to help people supporting our worst enemies get information they want, ''independent'' of controls that people with power exercise over nearly all major media today:  If our actions reduce the ability of their leaders to censor their media (and of our political and economic leaders to censor ours), the information everyone gets should make it harder for leaders to convince others to support measures contrary to nearly everyone's best interests.  

What kinds of data can we collect and analyze to evaluate who benefits and who loses from alternative interventions attempting to improve the media? See below.&lt;ref&gt;The power relationship between media and politicians can go both ways.  In addition to asking the extent to which politicians control the media, we can also consider the extent to which political leaders might feel constrained by the major media: To what extent do the major media create the stage upon which politicians read their lines, as claimed in the Wikiversity article on &quot;[[Confirmation bias and conflict]]&quot;?  Might a more diverse media environment make it easier for political leaders to pursue policies informed more by available research and less by propaganda?  Might experiments as described herein help politicians develop more effective governmental policies, because of a reduction in the power of media whose ownership and funding are more diverse? This is discussed further in this article in a section on [[Information is a public good: Designing experiments to improve government#Media and war|Media and war]].&lt;/ref&gt;

=== World Bank on the value of information === 

In 2002 the President of the [[w:World Bank|World Bank]],&lt;ref&gt;The 2022 World Bank Group portfolio was 104 billion USD (World Bank 2022, Table 1, p. 13; 17/116 in PDF).  An improvement of 0.1 percentage points in the performance of that portfolio would be 104 million.  A lot could be accomplished with budgets much smaller than this.&lt;/ref&gt; [[w:James Wolfensohn|James Wolfensohn]], wrote, &quot;[A] free press is not a luxury. It is at the core of equitable development. The media can expose corruption. ... They can facilitate trade [and bring] health and education information to remote villages ... . But ... the independence of the media can be fragile and easily compromised. All too often governments shackle the media. Sometimes control by powerful private interests restricts reporting. ... [T]o support development, media need the right environment{{mdash}}in terms of freedoms, capacities, and checks and balances.&quot;&lt;ref&gt;Wolfensohn (2002). More on this is available in other contributions to Islam et al. (2002) including [[w:Joseph Stiglitz|Stiglitz]] (2002), who noted the following: &quot;There is a natural asymmetry of information between those who govern and those whom they are supposed to serve. ... Free speech and a free press not only make abuses of governmental powers less likely, they also enhance the likelihood that people's basic social needs will be met. ... [S]ecrecy distorts the arena of politics. ... Neither theory nor evidence provides much support for the hypothesis that fuller and timelier disclosure and discussion would have adverse effects. ... The most important check against abuses is a competitive press that reflects a variety of interests. ... [F]or government officials to appropriate the information that they have access to for private gain ... is as much theft as stealing any other public property.&quot;&lt;/ref&gt;

=== US Postal Service Act of 1792: a natural experiment === 

[[w:Robert W. McChesney|McChesney]] and [[w:John Nichols (journalist)|Nichols]] (2010, 2016) suggested that the US [[w:Postal Service Act|Postal Service Act]]&lt;ref&gt;Wikipedia &quot;[[w:Postal Service Act|Postal Service Act]]&quot;, accessed 2023-07-11.&lt;/ref&gt; of 1792 made a major contribution to making the US what it is today.  Under that act, newspapers were delivered up to 100 miles for a penny, when first class postage was between 6 and 25 cents depending on distance.  They estimated that between 1840 and 1844, the US postal subsidy was 0.211 percent of GDP with federal printing subsidies adding another 0.005 percent, totaling 0.216 percent of GDP,&lt;ref name=McC-N2010&gt;McChesney and Nichols (2010, pp. 310-311, note 88).&lt;/ref&gt; roughly $140 per person per year in 2019 dollars.&lt;ref name=McN_IMF&gt;International Monetary Fund (2023):  US Gross domestic product per capita at current prices was estimated at $65,077 for 2019 on 2023-04-28. 0.211% of $65,077 = $137; round to $140 for convenience.&lt;/ref&gt; We use 2019 dollars here to make it easy to compare with Rolnik et al. (2019), who recommended $50 per adult per year, which is roughly 0.08 percent of US GDP.  Rolnik et al. added that the level of subsidies would require &quot;extensive deliberation and experimentation&quot;.&lt;ref name=Rolnik&gt;Rolnik et al. (2019, p. 44).  Per [[w:Demographics of the United States]], 24 percent of the US population is under 18, so adults are 76 percent of the population. Thus, $50 per adult is $37.50 per capita. US GDP per capita in $65,0077 in 2019 in current dollars per International Monetary Fund (2023). Thus, $37.50 per capita would be roughly 0.077 percent of GDP; round to 0.08 percent for convenience.&lt;/ref&gt; More recently McChesney and Nichols have recommended 0.15 percent of GDP, considering the fact that the advent of the Internet has nearly eliminated the costs of printing and distribution.&lt;ref name=McC-N2021&gt;McChesney and Nichols (2021; 2022, p. 19).&lt;/ref&gt;

[[w:Alexis de Tocqueville|Tocqueville]], who visited the US in 1831, observed the following:  
* [T]he liberty of the press does not affect political opinion alone, but extends to all the opinions of men, and modifies customs as well as laws. ... I approve of it from a consideration more of the evils it prevents, than of the advantages it insures.&lt;ref&gt;Tocqueville (1835; 2001, p. 91).  In 2002 Roumeen Islam stated this more forcefully:  &quot;Arbitrary actions by government are always to be feared. If there is to be a bias in the quantity of information that is released, then erring on the side of more freedom rather than less would appear to cause less harm.&quot; (World Bank, 2002, pp. 21-22; 33-34/336 in pdf).&lt;/ref&gt;
* The liberty of writing ... is most formidable when it is a novelty; for a people who have never been accustomed to hear state affairs discussed before them, place implicit confidence in the first tribune who presents himself.  The Anglo-Americans have enjoyed this liberty ever since the foundation of the Colonies ... .  A glance at a French and an American newspaper is sufficient to show the difference ... .  In France, the space allotted to commercial advertisements is very limited, and the news-intelligence is not considerable; but the essential part of the journal is the discussion of the politics of the day.  In America, three-quarters of the enormous sheet are filled with advertisements, and the remainder is frequently occupied by political intelligence or trivial anecdotes:  it is only from time to time that one finds a corner devoted to passionate discussions, like those which the journalists of France every day give to their readers.&lt;ref&gt;Tocqueville (1835; 2001, p. 92).&lt;/ref&gt; 
* It has been demonstrated by observation, and discovered by the sure instinct even of the pettiest despots, that the influence of a power is increased in proportion as its direction is centralized.&lt;ref&gt;Tocqueville (1835; 2001, pp. 92-93).&lt;/ref&gt;
* [T]he number of periodical and semi-periodical publications in the United States is almost incredibly large. In America there is scarcely a hamlet which does not have its newspaper.&lt;ref&gt;Tocqueville (1835; 2001, p. 93).&lt;/ref&gt;
* In the United States, each separate journal exercises but little authority; but the power of the periodical press is second only to that of the people ... .&lt;ref&gt;Tocqueville (1835; 2001, p. 94). &lt;/ref&gt;

[[File:Real US GDP per capita in 5 epocs.svg|thumb|Figure 1.  Average annual income (Gross Domestic Product per capita adjusted for inflation) in the US 1790-2021 showing five epochs identified in a &quot;breakpoint&quot; analysis (to 1929, 1933, 1945, 1947, 2021) documented in the Wikiversity article on &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;.&lt;ref&gt;Wikiversity &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;, accessed 2023-07-18.&lt;/ref&gt; Piketty (2021, p. 139) noted, &quot;In the United States, the national income per inhabitant rose at a rate ... of 2.2 percent between 1950 to 1990 when the top tax rate reached on average 72 percent.  The top rate was then cut in half, with the announced objective of boosting growth.  But in fact, growth fell by half, reaching 1.1 percent per annum between 1990 and 2020&quot;.&lt;ref&gt;A more recent review of the literature of the impact of inequality on growth is provided by Jahangir (2023, sec. 3), who notes that some studies have claimed that inequality ''increases'' the rate of economic growth, while other reach the opposite conclusion. However, 'the preponderant academic position is shifting from the argument that “we don’t have enough evidence” and towards seriously addressing and combating economic inequality.'&lt;/ref&gt; Our analysis of US GDP per capita from Measuring Worth do not match Piketty's report exactly, but they are close. We got 2.3 percent annual growth from 1947 to 1990 then 1.8 percent to 2008 and 1.1 percent to 2020. However, we have so far been unable to find a model that suggests that this decline is statistically significant.]]  

To what extent was [[w:Alexis de Tocqueville|Tocqueville's]] &quot;incredibly large&quot; &quot;number of periodical and semi-periodical publications in the United States&quot; due to the US Postal Service Act of 1792? To what extent did that &quot;incredibly large&quot; number of publications encourage literacy, limit political corruption, and help the US of that day remain together and grow both in land area and economically while contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically? To what extent does the enormous power of the US today rest on the economic growth of that period and its impact on the political culture of that day continuing to the present?&lt;ref&gt;Wikiversity &quot;[[The Great American Paradox]]&quot;, accessed 2023-06-12.&lt;/ref&gt; That growth transformed the US into the world leader that it is today;  see Figure 1.  In the process, it generated new technologies that benefit the vast majority of the world's population alive today. If the newspapers Tocqueville read made any substantive contribution to the growth summarized in Figure 1, the information in those newspapers were public goods potentially benefiting the vast majority of humanity ''to the end of human civilization.''&lt;ref&gt;Acemoglu (2023) documents how the power of monopolies and other politically favored groups often distorts the direction of technology development into suboptimal technologies.  Might increasing the funding for more independent news outlets reduce the power of such favored groups and thereby help correct these distortions and deliver &quot;sizable welfare benefits&quot;, e.g., &quot;in the context of industrial automation, health care, and energy&quot;?&lt;/ref&gt;

=== Other economists === 

We cannot prove that the diversity of newspapers in the early US contributed to the economic growth it experienced. Banerjee and Duflo (2019) concluded that no one knows how to create economic growth. They won the 2019 Nobel Memorial Prize in Economics with Michael Kremer for their leadership in using [[w:randomized controlled trials|randomized controlled trials]]&lt;ref&gt;Wikipedia &quot;[[w:Randomized controlled trials|Randomized controlled trials]]&quot;, accessed 2023-07-11.&lt;/ref&gt; to learn how to reduce global poverty.&lt;ref&gt;Wikipedia &quot;[[w:2019 Nobel Memorial Prize in Economic Sciences|2019 Nobel Memorial Prize in Economic Sciences]]&quot;, accessed 2023-06-13.  Nobel Prize (2019).  Amazon.com indicates that distribution of the book started 2019-11-12, twenty-nine days after the Nobel prize announcement 2019-10-14. Evidently the book must have been completed before the announcement.&lt;/ref&gt; More recently, Wake et al. (2021) found evidence that ''the economic costs of curbing press freedom persist long after such freedoms have been restored.''&lt;ref&gt;See also Nguyen et al. (2021).&lt;/ref&gt; And Mohammadi et al. (2022) found that economic growth rates were impacted by civil liberties, economic and press freedom and the economic growth rates of neighbors (spacial autocorrelation) but not democracy. These findings of Mohammidi et al. (2022) and Wake et al. (2021) reinforce Thomas Jefferson's 1787 comment that, &quot;were it left to me to decide whether we should have a government without newspapers, or newspapers without a government, I should not hesitate a moment to prefer the latter.&quot;&lt;ref&gt;From a letter to Colonel Edward Carrington (16 January 1787), cited in Wikiquote, &quot;[[Wikiquote:Thomas Jefferson|Thomas Jefferson]]&quot;, accessed 2023-07-29.&lt;/ref&gt; To what extent might experiments like those recommended in this article either reinforce or refute this claim of Jefferson from 1787?

=== Randomized controlled trials to quantify the value of information === 

This article suggests randomized controlled trials to quantify the impact of citizen-directed subsidies for journalism, roughly following the recommendations of McChesney and Nichols (2021, 2022) to distribute some small percentage of GDP to local news nonprofits ''via local elections''.  Philanthropies could fund such experiments for some of the smallest and poorest places in the world.  Organizations like the World Bank could fund such experiments as adjuncts to a random selection from some list of other interventions they fund, justified for the same reason that they would not consider funding anything without appropriate accounting and auditing of expenditures, as discussed further below.  

Before making suggestions regarding experiments, we review previous research documenting how information might be a public good.

== Previous research ==

Before considering optimal level of subsidies for news, it may be useful to consider the research for which [[w: Daniel Kahneman|Daniel Kahneman won the 2002 Nobel Memorial Prize in Economics]].&lt;ref&gt;Wikipedia &quot;[[w:Daniel Kahneman|Daniel Kahneman]]&quot;, accessed 2023-04-28.&lt;/ref&gt; Most important for present purposes may be that virtually everyone: 

# thinks they know more than they do ([[w:Overconfidence effect|Overconfidence]]),&lt;ref&gt;Wikipedia &quot;[[w:Overconfidence effect|Overconfidence effect]]&quot;, accessed 2023-04-29. Kahneman and co-workers have documented that experts are also subject to overconfidence and in some cases may be worse.  Kahneman and Klein (2009) found that expert intuition can be learned from frequent, rapid, high-quality feedback about the quality of their judgments.  Unfortunately, few fields have that much quality feedback.  Kaheman et al. (2021) call practitioners with credentials but without such expert intuition &quot;respect-experts&quot;.  Kahneman (2011, p. 234) said his &quot;most satisfying and productive adversarial collaboration was with Gary Klein&quot;.&lt;/ref&gt; and 
# prefers information and sources consistent with preconceptions. ([[w:Confirmation bias|Confirmation bias]]).&lt;ref&gt;Wikipedia &quot;[[w:Confirmation bias|Confirmation bias]]&quot;, accessed 2023-04-29.&lt;/ref&gt;

To what extent do media organizations everywhere exploit the confirmation bias and overconfidence of their audience to please those who control the money for the media, and to what extent might this ''reduce'' broadly shared economic growth?  The proposed experiments should include efforts to quantify this, measuring, e.g., local incomes, inequality, political polarization and the impact of interventions attempting to improve such.  

Plous wrote, &quot;No problem in judgment and decision making is more prevalent and more potentially catastrophic than overconfidence.&quot;&lt;ref&gt;Plous (1993, p. 217).  See also Wikipedia &quot;[[w:Overconfidence effect|Overconfidence effect]]&quot;, accessed 2023-04-29.&lt;/ref&gt; It contributes to inordinate losses by all parties in negotiations of all kinds&lt;ref&gt;Thompson (2020).&lt;/ref&gt; including lawsuits,&lt;ref&gt;Loftus and Wagenaar (1988).&lt;/ref&gt; strikes,&lt;ref&gt;Babcock and Olson (1992) and Thompson and Loevenstein (1992).&lt;/ref&gt; financial market bubbles and crashes,&lt;ref&gt;Daniel et al. (1998).&lt;/ref&gt; and politics and international relations,&lt;ref&gt;&lt;!-- Dominic D.P. Johnson (2020) Strategic instincts: the adaptive advantages of cognitive biases in international politics--&gt;{{cite Q|Q120967807}}&lt;/ref&gt; including wars.&lt;ref&gt;Johnson (2004).&lt;/ref&gt;

Might the frequency and expense of lawsuits, strikes, financial market volatility, political coruption and wars be reduced by encouraging people to get more curious and search more often for information that might contradict their preconceptions?  Might such discussions be encouraged by interventions such as increasing the total funding for news through many small, independent, local news organizations?  If yes, to what extent might such experimental interventions threaten the hegemony of major media everywhere while benefitting everyone, with the possible exception of those who benefit from current systems of political corruption?  

[[File:Knowledge v. public media.png|thumb|Figure 2.  Knowledge v. public media: Percent correct answers in surveys of knowledge of domestic and international politics vs. per capita subsidies for public media in Denmark (DK), Finland (FI), the United Kingdom (UK) and the United States (US).&lt;ref&gt;&quot;politicalKnowledge&quot; dataset in Croissant and Graves (2022), originally from ch. 1, chart 8, p. 268 and ch. 4, chart 1, p. 274, McChesney and Nichols (2010).&lt;/ref&gt;]]

One attempt to quantify this appears in Figure 2, which summarizes a natural experiment on the impact of government subsidies for public media on public knowledge of domestic and international politics: Around 2008 the governments of the US, UK, Denmark and Finland provided subsidies of $1.35, $80, $101 and $101 per person per year, respectively, for public media.  A survey of public knowledge of domestic and international politics found that people with college degrees seemed to be comparably well informed in the different countries, but people with less education were better informed in the countries with higher public subsidies.  

Kaviani et al. (2022) studied the impact of &quot;the staggered expansion of [[w:Sinclair Broadcast Group|Sinclair Broadcast Group]]: the largest conservative network in the U.S.&quot; They documented a decline in Corporate Social Responsibility (CSR) ratings of firms headquartered in Sinclair expansion areas.  They also documented a &quot;right-ward ideological shift&quot; in coverage that was &quot;nearly one standard deviation of the ideology distribution&quot; as well as &quot;substantial decreases in coverage of local politics substituted by increases in national politics.&quot; Ellison (2024) said that &quot;Sinclair's recipe for TV news&quot; includes an annual survey asking viewers, &quot;What are you most afraid of?&quot; Sinclair reportedly focuses on that while implying in their coverage &quot;that America's cities, especially those run by Democratic politicians, are dangerous and dysfunctional.&quot; 

Sources in France are concerned that billionaire [[w:Vincent Bolloré|Vincent Bolloré]] has purchased a substantial portion of French media and used it effectively to promote the French far right.&lt;ref&gt;Francois (2022). Cagé (2022). Cagé and Stetler (2022).&lt;/ref&gt; Scheidler (2024a) reported that the concentration of ownership the German media &quot;has not yet reached the extreme forms observed in France, the United Kingdom or the United States, but the process of consolidation initiated several decades ago has transformed a landscape renowned for its decentralization.&quot;&lt;ref&gt;Translated from, &quot;la concentration de la propriété dans la presse suprarégionale n’a pas encore atteint les formes extrêmes observées en France, au Royaume-Uni ou aux États-Unis, mais le processus de consolidation enclenché depuis plusieurs décennies a transformé un paysage réputé pour sa décentralisation.&quot;&lt;/ref&gt;&lt;ref&gt;See also ''Die Tageszeitung'' (2023).&lt;/ref&gt; Scheidler (2024b) reported that there still exists a wide range of constructive media criticism in Germany, but it gets less coverage than before in the increasingly consolidated major media. This has driven many who are not happy with these changes to alternative media such as ''[[w:Die Tageszeitung|Die Tageszeitung]]'', founded in 1978.

Benton wrote that past research has shown that strong local newspapers &quot;increase voter turnout, reduce government corruption, make cities financially healthier, make citizens more knowledgeable about politics and more likely to engage with local government, force local TV to raise its game, encourage split-ticket (and thus less uniformly partisan) voting, make elected officials more responsive and efficient ... And ... you get to reap the benefits of all those positive outcomes ''even if you don’t read them yourself''.&quot;&lt;ref&gt;Benton (2019); italics in the original.  See also Green et al. (2023, p. 7), Schulhofer-Wohl and Garrido (2009) and Stearns and Schmidt (2022).  A not quite silly example of this is documented in the Wikipedia article on the &quot;[[w:City of Bell scandal|City of Bell scandal]]&quot; accessed 2023-05-05:  Around 1999 the local newspaper died.  In 2010 the ''[[w:Los Angeles Times|Los Angeles Times]]'' reported that the city was close to bankruptcy in spite of having atypically high property tax rates.  The compensation for the City Manager was almost four times that of the President of the US, even though Bell, California, had a population of only approximately 38,000.  The Chief of Police and most members of the City Council also had exceptionally high compensations.  It was as if the City Manager had said in 1999, &quot;Wow:  The watchdog is dead.  Let's have a party.&quot;&lt;/ref&gt;

We feel a need to repeat that last comment:  ''You and I'' benefit from others consuming news that we do not, because they become less likely to be stampeded into voting contrary to their best interests{{mdash}}and ours{{mdash}}and more likely to lobby effectively against questionable favors to major political campaign contributors or other people with power, underreported by major media that have conflicts of interest in balanced coverage of anything that might offend people with substantive control of their funding.  That suggests that everyone might benefit from subsidizing ''a broad variety of independent'' local news outlets consumed by others.&lt;ref&gt;Some of those who benefit from the current system of political corruption may lose from the increased transparency produced by increases in the quality, quantity, diversity, and broader consumption of news.  However, Bezruchka (2023) documents how even the ultra-wealthy in countries with high inequality generally have shorter life expectancies than their counterparts in more egalitarian societies:  What they might lose in social status would likely be balanced by a reduction in stress and exposure to life-threatening incidents.&lt;/ref&gt; Experiments along the lines discussed below could attempt to evaluate these claims and estimate their magnitudes.

== How fair is the US tax system? ==

How fair is the US tax system?  It depends on who is asked and how fairness is defined.  

[[File:Share of taxes vs. AGI.svg|thumb|Figure 3.  Effective tax rate vs. Adjusted Gross Income (AGI).&lt;ref&gt;York (2023) based on analyses published by the US Internal Revenue Service (IRS).&lt;/ref&gt;]]

The [[w:Tax Foundation|Tax Foundation]] computed the effective tax rate in different portions of the distribution of Adjusted Gross Income (AGI), plotted in Figure 3.  They noted that,&quot;half of taxpayers paid 99.7 percent of federal income taxes&quot;. The effective tax rate on the 1 percent highest adjusted gross incomes (AGIs) was 26 percent, almost double (1.91 times) the average, while the effective tax rate for the bottom half was 3.1 percent, only 23 percent of the average.&lt;ref&gt;York (2023).&lt;/ref&gt;

The Tax Foundation did ''not'' mention that we get a very different perspective from considering ''gross income'' rather than AGI. Leiserson and Yagan (2021)&lt;ref&gt;published by the Biden White House.&lt;/ref&gt; estimated that the average ''effective'' federal individual income tax rate paid by America’s 400 wealthiest families&lt;ref&gt;The &quot;400 wealthiest families&quot; are identified in &quot;[[w:The Forbes 400|The Forbes 400]]&quot;; see the Wikipedia article with that title, accessed 2023-05-07.&lt;/ref&gt; was between 6 and 12 percent with the most likely number being 8.2 percent. The difference comes in the ''adjustments'', while the uncertainty comes primarily from appreciation in the value of unsold stock,&lt;ref&gt;Unsold stock or other property subject to capital gains tax, which in 2022 was capped at 20 percent;  see Wikipedia, &quot;[[w:Capital gains tax in the United States|Capital gains tax in the United States]]&quot;, accessed 2023-05-08.&lt;/ref&gt; which is taxed at a maximum of 20 percent when sold and never taxed if passed as inheritance.&lt;ref&gt;The Wikipedia article on &quot;[[w:Estate tax in the United States|Estate tax in the United States]]&quot; describes an &quot;Exclusion amount&quot;, which is not taxed in inheritance.  That exclusion amount was $675,000 in 2001 and has generally trended upwards since except for 2010, and was $12.06 million in 2022. (accessed 2023-05-08.)&lt;/ref&gt;

Divergent claims about ''business'' taxes can similarly be found.  Watson (2022) claimed  that, &quot;Corporate taxes are one of the most economically damaging ways to raise revenue and are a promising area of reform for states to increase competitiveness and promote economic growth, benefiting both companies and workers.&quot;  His &quot;economically damaging&quot; claim seems contradicted by the claim of Piketty (2021, p. 139), cited with Figure 1 above, that when the top income tax rate was cut in half, the rate of economic growth in the US fell by half, instead of increasing as Watson (2022) suggested. By contrast, Fuhrmann and Uradu (2023) describe, &quot;How large corporations avoid paying taxes&quot;.   

[[File:UStaxWords.svg|thumb|Figure 4.  Millions of words in the US federal tax code and regulations, 1955-2015, according to the [[w:Tax Foundation|Tax Foundation]]. [1=income tax code; 2=other tax code; 3=income tax regulations; 4=other tax regulations; solid line= total]&lt;ref&gt;&quot;UStaxWords&quot; dataset in Croissant and Graves (2022) from the Tax Foundation.&lt;/ref&gt;]]

One reference on the difference between &quot;adjusted&quot; and &quot;gross&quot; income is US federal tax code and regulations, which grew from 1.4 million words in 1955 to over 10 million in 2015, averaging 145,000 additional words each year;  see Figure 4.  

How does this relate to media?

== How do media organizations make money? ==

Media organizations everywhere sell changes in audience behaviors to the people who give them money.  If they do not have an audience, they have nothing to sell.  If they sufficiently offend their funders, they will not get the revenue needed to produce content.&lt;ref&gt;A famous illustration of this conflict between content and funding was when CBS Chairman [[w:William S. Paley|William Paley]] reportedly told [[w:Edward R. Murrow|Edward R. Murrow]] in 1958 that he was discontinuing Murrow's award-winning show ''[[w:See It Now|See It Now]]'', because &quot;I don't want this constant stomach ache every time you do a controversial subject&quot;, documented in Friendly (1967, p. 92).&lt;/ref&gt;  

The major media in the US have conflicts of interest in honestly reporting on discussions in congress on copyright law or on anything that might impact a major advertiser or might make it easier for politicians to get elected by spending less money on advertising.  McChesney (2015) insisted that the major media are not interested in providing information that people want:  They are interested in making money and protecting the interests of the ultra-wealthy, who control the largest advertising budgets.  For example, media coverage of the roughly 40,000 people who came to [[w:1999 Seattle WTO protests|Seattle in 1999 to protest the WTO]] Ministerial Conference there&lt;ref&gt;Wikipedia &quot;[[w:1999 Seattle WTO protests|1999 Seattle WTO protests]]&quot;, accessed 2023-05-08.&lt;/ref&gt; and the 10,000 - 15,000 who came to [[w:Washington A16, 2000|Washington, DC, the following year]] to protest the International Monetary Fund and the World Bank,&lt;ref&gt;Wikipedia &quot;[[w:Washington A16, 2000|Washington A16, 2000]]&quot;, accessed 2023-05-08.&lt;/ref&gt; included &quot;some outstanding pieces produced by the corporate media, but those were exceptions to the rule. ... [T]he closer a story gets to corporate power and corporate domination of our society, the less reliable the corporate news media is.&quot;&lt;ref&gt;McChesney (2015, p. xx).&lt;/ref&gt; 

Aaron (2021) said, &quot;Bob McChesney ... taught me [to] look at ... the stories that are cheap to cover.&quot; Between around 1975 and 2000, the major commercial broadcasters in the US fired nearly all their investigative journalists&lt;ref&gt;McChesney (2004, p. 81):  &quot;A five-year study of investigative journalism on TV news completed in 2002 determined that investigative journalism has all but disappeared from the nation's commercial airwaves.&quot;&lt;/ref&gt; and replaced them with the police blotter.  It's easy and cheap to repeat what the police say.&lt;ref&gt;Holmes (2022) quoted Ryan Sorrell, Founder and Publisher of the Kansas City Defender, as saying, &quot;the media often parrots or repeats what police and news releases say.&quot;&lt;/ref&gt; A news outlet can do that without seriously risking loss of revenue.  In addition, poor defendants who may not have money for legal defense will rarely have money to sue a media outlet for defamation.  By contrast, a news report on questionable activities by a major funder risks both direct loss of advertising revenue and being sued.&lt;ref&gt;The risks of being sued include the risks of [[w:Strategic lawsuit against public participation|strategic lawsuits against public participation]] (SLAPPs) by major organizations, which can intimidate journalists and publishers as well as potential whistleblowers, who might inform journalists of violations of law by their employers.  Some of these are documented in the &quot;[[w:Freedom of the press in the United States#U.S. Press Freedom Tracker|U.S. Press Freedom Tracker]]&quot;, maintained by the [[w:Freedom of the Press Foundation|Freedom of the Press Foundation]] and the [[w:Committee to Protect Journalists|Committee to Protect Journalists]].  These include arrests, assaults, threats, denial of access, equipment damage, prior restraint, and subpoenas which could intimidate journalists, publishers, and employees feeling a need to expose violations of law and threats to public safety.  See Wikipedia &quot;[[w:Freedom of the Press Foundation|Freedom of the Press Foundation]]&quot;, &quot;[[w:Committee to Protect Journalists|Committee to Protect Journalists]]&quot;, and &quot;[[w:Strategic lawsuit against public participation|Strategic lawsuit against public participation]]&quot;, accessed 2023-07-11.&lt;/ref&gt;  These risks impose a higher standard of journalism (and additional costs) when reporting on questionable activities by people with power than when reporting on poor people.  This is a much bigger problem in countries where libel is a criminal rather than a civil offense or where truth is not a defense for libel.&lt;ref&gt;Islam et al. (2002), esp. pp. 12-13 (24-25/336 in pdf), p. 50 (62/336 in pdf), and ch. 11, pp. 207-224 (219-236/336 in pdf). [[w:United States defamation law|Truth was not a defense against libel in the US]] in 1804 when Harry Croswell lost in ''[[w:United States defamation law#People v. Croswell|People v. Croswell]]''.  That began to change the next year when the [[w:United States defamation law#People v. Croswell|New York State Legislature]] changed the law to allow truth as a defense against a libel charge. Seventy years earlier in 1735 [[w:John Peter Zenger#Libel case|John Peter Zenger]] was acquitted of a libel charge, but only by [[w:Jury nullification in the United States|jury nullification]].&lt;/ref&gt;

[[File:U.S. incarceration rate since 1925.svg|thumb|Figure 5.  Percent of the US population in state and federal prisons  [male (dashed red), combined (solid black), female (dotted green)]&lt;ref&gt;&quot;USincarcerations&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

After about 1975 the public noticed the increased coverage of crime in the broadcast news and concluded that crime was out of control, when there had been no substantive change in crime.  They voted in a generation of politicians, who promised to get tough on crime.  The incarceration rate in the US went from 0.1 percent to 0.5 percent in the span of roughly 25 years, after having been fairly stable for the previous 50 years;  see Figure 5.&lt;ref&gt;Potter and Kapeller (1998). Sacco (1998, 2005).&lt;/ref&gt;  

[[File:IncomeInequality9b.svg|thumb|Figure 6.  Average and quantiles of family income (Gross Domestic Product per family) in constant 2010 dollars.&lt;ref&gt;&quot;incomeInequality&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

Around that same time, income inequality in the US began to rise;  see Figure 6.&lt;ref&gt;Bezruchka (2023) summarizes research documenting how &quot;inequality kills us all&quot;. He noted that the US was among the leaders in infant mortality and life expectancy in the 1950s.  Now the US is trailing most of the advanced industrial democracies per United Nations (2022).  He attributes the slow rate of improvement in public health in the US to increases in inequality.  That argument is less than perfect, because Figure 5 suggests that inequality in the US did not begin to increase until around 1975, but the divergence in public health between the US and other advanced industrial democracies seems more continuous between the 1950s and the present, 2023.  Beyond this, Graves and Samuelson (2022) noted that it is in everyone's best interest to help others with conditions that might be infectious to get competent medical assistance, because that reduces our risk of contracting their disease and possibly dying from it. Bezruchka (2023) cites documentation claiming that even the wealthy in the US have lower life expectancy than their counterparts in other advanced industrial democracies, because the high level of inequality in the US means that the ultra-wealthy in the US get exposed to more pathogens than their counterparts elsewhere.  See also Wilkinson and PIckett (2017).&lt;/ref&gt; To what extent might that increase in inequality be due to the structure of the major media?  

To what extent might you and I benefit from making it easier for millions of others to research different aspects of government policies including the &quot;adjustments&quot; in the US tax system embedded in the over 10 million words of US federal tax code and regulations documented with Figure 4 above, encouraging them to lobby the US Congress against the special favors granted to major political campaign contributors against the general welfare of everyone else? Everyone except the beneficiaries of such political corruption would likely benefit from the news that helps concerned citizens lobby effectively against such corruption, even if we did not participate in such citizen lobbying efforts and were completely ignorant of them.

== Media and war == 

:[[w:You've Got to Be Carefully Taught|''&quot;You've got to be taught to hate and fear. ... It's got to be drummed in your dear little ear.&quot;'']] 
:Lt. Cable in the [[w:South Pacific (musical)|1949 Rodgers and Hammerstein musical ''South Pacific'']].  

To what extent is it accurate to say that before anyone is killed in armed hostilities, the different parties to the conflict are polarized by the different media the different parties find credible?&lt;ref&gt;The role of the media in war has long been recognized.  It is commonly said that the first casualty of war is truth. Knightley (2004, p. vii) credits Senator Hiram Johnson as saying in 1917, &quot;The first casualty when war comes is truth.&quot; However, the Wikiquote article on &quot;[[Wikiquote:Hiram Johnson|Hiram Johnson]]&quot; says this quote has been, &quot;Widely attributed to Johnson, but without any confirmed citations of original source. ... [T]he first recorded use seems to be by Philip Snowden.&quot; (accessed 2023-07-22.)&lt;/ref&gt; 

This might seem obvious, but how can we quantify political polarization in a way (a) that correlates with the severity of the conflict and (b) can be used to evaluate the effectiveness of efforts to reduce the polarization? 

The [[w:International Panel on the Information Environment|International Panel on the Information Environment]] (IPIE) is a consortium of over 250 global experts developing tools to combat political polarization driven by the structure of the Internet.&lt;ref&gt;e.g., National Acadamies (2023).&lt;/ref&gt; The US Institute of Peace (2016) discusses &quot;Tools for Improving Media Interventions in Conflict Zones&quot;. Previous research in this area was summarized by Arsenault et al. (2011). One such tool might be video games.&lt;ref&gt;Caelin (2016).&lt;/ref&gt;

We suggest experimenting with interventions designed to reduce political polarization with some of the smallest but most intense conflicts: Interventions that require money could be more effectively tested with smaller, high intensity conflicts. With randomized controlled trials, it would be easier to measure a reduction from a higher-intensity conflict, and a smaller population could commit more money per capita with a relatively modest budget.

The [[w:Armed Conflict Location and Event Data Project|Armed Conflict Location and Event Data Project (ACLED)]] tracks politically relevant violent and nonviolent events by a range of state and non-state actors. Their data can help identify countries or geographic regions in conflict as candidates to be [[w:Randomized controlled trial|assigned randomly to experimental and control groups]], whose comparison can provide high quality data to help evaluate the impact of any intervention. Initial experiments of this nature might be done with a modest budget by working with organizations advocating nonviolence and with religious groups to recruit diaspora communities to do things recommended by experts in IPIE while also lobbying governments for funding. Any success can be leveraged into changes in foreign and military policies to make the world safer for all. 

Before discussing such experiments further, we consider a few case studies. 

=== Russo-Ukrainian War and the US Civil War === 

In the [[w:Russo-Ukrainian War|Russo-Ukrainian War]], Halimi and Rimbert (2023) describe &quot;Western media as cheerleaders for war&quot;. [[w:Joseph Stiglitz|Stiglitz]] (2002) noted this was a general phenomenon:  &quot;In periods of perceived conflict ... a combination of self-censorship and reader censorship may also undermine the ability of a supposedly free press to ensure democratic transparency and openness.&quot; Media organizations do not always do this solely to please their funders.  Reporters are killed&lt;ref&gt;Different lists of journalists killed for their work are maintained by the [[w:Committee to Protect Journalists|Committee to Protect Journalists]], (CPJ), [[w:Reporters Without Borders|Reporters without Borders]], and the [[w:International Federation of Journalists|International Federation of Journalists]].  CPJ has claimed that their numbers are typically lower, because their confirmation process may be more rigorous.  See Committee to Protect Journalists (undated) and the Wikipedia articles on &quot;[[w:Committee to Protect Journalists|Committee to Protect Journalists]]&quot;, &quot;[[w:Reporters Without Borders|Reporters without Borders]]&quot;, and the &quot;[[w:International Federation of Journalists|International Federation of Journalists]]&quot;, accessed 2023-07-11.&lt;/ref&gt; or jailed and news outlets closed to prevent them from disseminating information that people with power do not want distributed.  Early in the Civil War in the US (1861-1865), some newspapers in the North said the US should let the South secede, because that would be preferable to war.  Angry mobs destroyed some offices and printing presses.  One editor &quot;was forcibly taken from his house by an excited mob, ... covered with a coat of tar and feathers, and ridden on a rail through the town.&quot; Others changed their policies &quot;voluntarily&quot;, recognizing threats to their lives or property or to a loss of audience.&lt;ref&gt;Harris (1999, esp. p. 100).&lt;/ref&gt;

=== Hitler ===

Fulda (2009) studied the co-evolution of newspapers and party politics in Germany, focusing primarily on Berlin, 1924-1933. During that period, the [[w:Nazi Party|Nazis (NSDAP, National Socialist German Workers' Party, Nationalsozialistische Deutsche Arbeiterpartei)]] grew from 2.6 percent of the votes for the [[w:Reichstag (Nazi Germany)|Reichstag (German parliament)]] in 1928 to 44 percent in 1933. Fulda described exaggerations in the largely tabloid press of an indecisive government incapable of managing either the economy or the increasing political violence, blamed excessively on Communists, and the potential for civil war. This turned the Nazis into an attractive choice for voters desperate for decisive action.&lt;ref&gt;Fulda (2009, Abstract, ch. 6, &quot;War of Words: The Spectre of Civil War, 1931–2&quot;.&lt;/ref&gt; 

After the 1933 elections, the Reichstag passed the &quot;[[w:Enabling Act of 1933|Enabling Act of 1933]], which gave Hitler's cabinet the right to enact laws without the consent of parliament. The Nazis then began full censorship of the newspapers, physically beating, imprisoning and in some cases killing journalists, as the leading publishers acquiesced. The primary sources of news during that period was newspapers; German radio was relatively new during that period and carried very little news. Most newspapers were [[w:Tabloid journalism|tabloids]], interested in either making money or promoting a party line with minimal regard for fact checking. A big loser in this was the right‐wing press magnate [[w:Alfred Hugenberg|Alfred Hugenberg]], whose political mismanagement led to the substantial demise of his [[w:German National People's Party|German National People's Party (Deutschnationale Volkspartei, DNVP)]], mostly benefitting Hitler.&lt;ref&gt;Fulda (2009).&lt;/ref&gt; 

This suggests the need for a [[w:Counterfactual history|counterfactual analysis of this period]], asking what kinds of changes in the structure of the media ecology might have prevented the rise of the Nazis? In particular, to what extent might a more diverse local news environment supported by citizen-directed subsidies as suggested herein have reduced the risk of a demise of democracy? And might some sort of [[w:Fairness Doctrine|fairness doctrine]] have helped?&lt;ref name=fairness&gt;Wikipedia &quot;[[w:FCC fairness doctrine|FCC fairness doctrine]]&quot;, accessed 2023-07-21.&lt;/ref&gt; And how might different rules for distributing different levels of funding to local news outlets impact the level of democratization? (Threats to democracy include legislation like the German Enabling Act of 1933 and other situations that allow an executive to successfully ignore the will of an otherwise democratic legislature, a [[w:self-coup|self-coup]], as well as a military coup.)

=== Stalin and Putin === 

[[File:Russian economic history 1885-2018.svg|thumb|Figure 7. Gross Domestic Product per person in Russian 1885-2018 in thousands of 2011 dollars]]

A 2017 survey asking Russians to name 10 of the world’s most prominent personalities listed Joe Stalin and Vladimir Putin as the top two with 38 and 34 percent, respectively. When the study was redone in 2021, Putin had slipped from number 2 to number 5. Stalin still led with 39 percent followed by Vladimir Lenin with 30 percent, Poet Alexander Pushkin and tsar Peter the Great with 23 and 19 percent each, then Putin with 15 percent.&lt;ref&gt;&lt;!-- Putin Plummets, Stalin Stays on Top in Russians’ Ranking of ‘Notable’ Historical Figures – Poll--&gt;{{cite Q|Q123197680}} &lt;!-- The most outstanding personalities in history (in Russian) --&gt;{{cite Q|Q123197317}}&lt;/ref&gt; 

It may be difficult for some people in the West to understanding how Stalin and Putin could be so popular, given the way they have been typically described in the mainstream Western media.&lt;ref&gt;[[w:Joseph Stalin|Joseph Stalin]] got much better press in the US during the Great Depression and World War II than he has gotten since 1945.&lt;/ref&gt; However, this is relatively easy to understand just by looking at the accompanying plot (Figure 7) of average annual income in that part of the world between 1885 and 2018: Both Stalin and Putin inherited economies that had fallen dramatically in the previous years and supervised dramatic improvements. Putin's decline between 2017 and 2021 may also be understood from this plot, because it shows how the dramatic growth that began around the time that Putin became acting President of Russia has slowed substantially since 2012. 

Similar comments could be made about the Vietnam war and the &quot;War on Terror&quot;.&lt;ref&gt;The Wikiversity article on &quot;[[Winning the War on Terror]]&quot; discusses the role of the media in the &quot;War on Terror&quot; and other conflicts including Vietnam.&lt;/ref&gt; To what extent can the experiments described in this article contribute to understanding the role of the media in stoking hate and fear, and how that might be impacted by citizen-direct subsidies for more and more diverse local media?

=== Iraq and the Islamic State ===

[[w:Fall of Mosul|In 2014 in Mosul, two Iraqi army divisions totaling 30,000 and another 30,000 federal police]] were overwhelmed in six days by roughly 1,500 committed Jihadists. Four months later, ''Reuters'' reported that, &quot;there were supposed to be close to 25,000 soldiers and police in the city; the reality ... was at best 10,000.&quot; Many of the missing 15,000 were &quot;ghost soldiers&quot; kicking back half their salaries to their officers. Also, &quot;[i]nfantry, armor and tanks had been shifted to Anbar, where more than 6,000 soldiers had been killed and another 12,000 had deserted.&quot;&lt;ref&gt;Parker et al. (2014).&lt;/ref&gt; 

To what extent might the political corruption and low moral documented in that ''Reuters'' report have been allowed to grow to that magnitude if Iraq had had a vigorous adversarial press, as discussed in this article? Instead, Paul Bremer, who was appointed as the [[w:Paul Bremer#Provisional coalition administrator of Iraq|Provisional coalition administrator of Iraq]] just over a week after President George W. Bush's [[w:Mission Accomplished speech|Mission Accomplished speech]] of 2003-05-01, imposed strict press censorship.&lt;ref&gt;McChesney and Nichols (2010, p. 242).&lt;/ref&gt; McChesney and Nichols contrasted this with General Eisenhower, who &quot;called in German reporters [after the official surrender of Nazi Germany in WW II] and told them he wanted a free press. If he made decisions that they disagreed with, he wanted them to say so in print.&quot;&lt;ref&gt;McChesney and Nichols (2010, Appendix II. Ike, MacArthur and the Forging of Free and Independent Press, pp. 241-254).&lt;/ref&gt;

=== Israel-Palestine ===

::''Those who make peaceful revolution impossible will make violent revolution inevitable.''
:::-- John F. Kennedy (1962)

To what extent is the [[w:Israeli–Palestinian conflict|Israeli–Palestinian conflict]] driven by differences in the media consumed by the different parties to that conflict? 

* To what extent are the supporters of Israel aware of violent acts committed by Palestinians but are ''unaware'' of the actions by Israelis that have motivated those violent acts? 
* Similarly, to what extent are the supporters of Palestinians unaware of or downplay the extent to which violence by Palestinians motivate the actions of Israel against them? 

To what extent are these differences in perceptions between supporters of Israel and supporters of Palestinians driven by differences in the media each find credible? 

What can be done to bridge this gap?  

[[w:Gene Sharp|Gene Sharp]], [[w:Mubarak Awad|Mubarak Awad]], and other supporters of [[w:nonviolence|nonviolence]] have suggested that when nonviolent direct action works, it does so by exposing a gap between the rhetoric [supported by the major media] and the reality of their opposition. Over time, this gap erodes pillars of support of the opposition. One example was the nonviolence of the [[w:First Intifada|First Intifada]] (1987-1993), which were protests against &quot;beatings, shootings, killings, house demolitions, uprooting of trees, deportations, extended imprisonments, and detentions without trial.&quot;&lt;ref&gt;Ackerman and DuVall (2000, p. 407).&lt;/ref&gt; As that campaign began, Israel got so much negative press for killing nonviolent protestors, that Israeli Defense Minister [[w:Yitzhak Rabin|Yitzhak Rabin]] ordered his soldiers NOT to kill but instead to shoot to wound. As the negative press continued, he issued wooden and metal clubs with orders to break bones.&lt;ref&gt;Shlaim (2014).&lt;/ref&gt; As the negative press still continued, Rabin ran for Prime Minister on a platform of negotiating with Palestinians. His victory and subsequent negotiations led to the [[w:Oslo Accords|Oslo Accords]] and the joint recognition of each other by the states of Israel and [[w:State of Palestine|Palestine]]. The West Bank and Gaza have continued under Israeli occupation since with some services provided by the official government of Palestine. 

During the Intifada, Israel tried to infiltrate the protestors with [[w:agent provocateur|agents provocateurs]] in Palestinian garb. They were exposed and neutralized until Israel deported 481 people they thought were leading the nonviolence who were accepted in other countries and imprisoned tens of thousands of others suspected of organizing the nonviolence. Finally, they got the violence needed to justify a massively violent repression of the Intifada.&lt;ref&gt;King (2007).&lt;/ref&gt;

The general thrust of this current analysis suggests a two pronged intervention to reduce the risk of a continuation of the violence that has marked Israel-Palestine since at least 1948: 
# Offer nonviolence training to all Palestininans, Israelis and supporters of either interested in the topic. This is the opposite of the policies Israel pursued during the First Intifada, at least according to the references cited in this discussion of that campaign.&lt;ref&gt;It also is the opposite of the decision of the US Supreme Court in ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', which ruled that teaching nonviolence to someone designated as a terrorist was a crime under the [[w:Patriot Act|Patriot Act]], as it provided &quot;material support to&quot; a foreign terrorist organization.&lt;/ref&gt;
# Provide citizen-direct subsidies to local news nonprofits in the West Bank and Gaza at, e.g., 0.15 percent of GDP, as recommended by McChesney and Nichols, cited above. 

How can we evaluate the budget required for such an experiment? The nominal GDP of the [[w:State of Palestine|State of Palestine]] in 2021 was estimated at $18 billion; 0.15% of that is $27 million. Add 10% for research to get $30 million per year. That ''annual'' cost for the media component of this proposed intervention is 12% of the billion Israeli sheckels ($246 million) that the Gaza war was costing Israel ''each day'' in the early days of the [[w:Israel-Hamas war|Israel-Hamas war]], according to the Israeli Finance Minister on 2023-10-25.&lt;ref&gt;Reuters (2023-10-25).&lt;/ref&gt; As this is being written, that war has continued for over 100 days. If the average daily cost of that war to Israel during that period has been $246 million, then that war will have already cost Israel over $24.6 billion. And that does not count the loss of lives and the destruction of property in Gaza and the West Bank. 

How much would training in nonviolence cost? That question would require more research, but if it were effective, the budget would seem to be quite modest compared to the cost of war, even if it were several times the budget for citizen-directed subsidies for local news in Palestine as just suggested.

== The decline of newspapers == 

[[File:Newspapers as a percent of US GDP.svg|thumb|Figure 8.  US newspaper revenue 1955-2020 as a percent of GDP.&lt;ref&gt;&quot;USnewspapers&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

McChesney and Nichols (2022) noted that US newspaper revenue as a percent of GDP fell from over 1 percent in 1956 to less than 0.1 percent in 2020;  see Figure 8.  Abernathy (2020) noted that the US lost more than half of all newspaper journalists between 2008 and 2018.&lt;ref&gt;Abernathy (2020, p. 22).&lt;/ref&gt; A quarter of US newspapers closed between 2004 and 2020,&lt;ref&gt;Abernathy (2020, p. 21).&lt;/ref&gt; and many that still survive are publishing less, creating &quot;news deserts&quot; and &quot;ghost newspapers&quot;, some with no local journalists on staff.&lt;ref&gt;Abernathy (2020) documented the problem of increasing &quot;news deserts and ghost newspapers&quot; in the US. A local jurisdiction without a local news outlet has been called a &quot;news desert&quot;.  She uses the term &quot;ghost newspapers&quot; to describe outlets &quot;with depleted newsrooms that are only a shadow of their former selves.&quot;  Some “ghost newspapers” continue to publish with zero local journalists, produced by reporters and editors that don't live there. One example is the ''Salinas Californian'', a 125-year-old newspaper in Salinas, California, which lost its last paid journalist 2022-12, according to the Los Angeles Times (2023-03-27).  They continue to publish, though &quot;The only original content from Salinas comes in the form of paid obituaries, making death virtually the only sign of life at an institution once considered a must-read by many Salinans.&quot; A leading profiteer in this downward spiral is reportedly [[w:hedge fund|hedge fund]] [[w:Alden Global Capital|Alden Global Capital]].  Threisman (2021) reported that, &quot;When this hedge fund buys local newspapers, democracy suffers&quot;.  And Benton (2021) said, &quot;The vulture is hungry again: Alden Global Capital wants to buy a few hundred more newspapers&quot;. Hightower (2023) describes two organizations fighting this trend.  One is National Trust for Local News, a nonprofit that recently bought several local papers and &quot;is turning each publication over to local non-profit owners and helping them find ways to become sustainable.&quot; The other is CherryRoad Media, which &quot;bought 77 rural papers in 17 states, most from the predatory Gannett conglomerate that wanted to dump them&quot;, and is working to &quot;return editorial decision-making to local people and journalists ... and ... reinvest profits in real local journalism that advances democracy.&quot; News outlets acquired by something like the National Trust for Local News should be eligible for citizen-directed subsidies for local news, as discussed below, after their ownership was officially transferred to local humans.  Outlets acquired by organizations like CheeryRoad Media would not be eligible as long as they remained subsidiaries.&lt;/ref&gt; More recent news continues to be dire. The Fall 2023 issue of ''Columbia Journalism Review'' reported that 2023 &quot;has become media’s worst year on record for job losses&quot;.&lt;ref&gt;Columbia Journalism Review (2023).&lt;/ref&gt; Substantial advertising revenue has shifted to the &quot;click economy&quot;, where advertisers pay for clicks, especially on social media.&lt;ref&gt;Carter (2021).&lt;/ref&gt;

Newpapers in other parts of the world have also experienced substantial declines in revenue. In 2013 German law was changed to inclued &quot;[[w:Ancillary copyright for press publishers|Ancillary copyright for press publishers]]&quot;, also called a &quot;link tax&quot;. However, this law was declared invalid in 2019 the European Court of Justice (ECJ), because it had been submitted in advance to the [[w:European Commission|EU Commission]], as required.&lt;ref&gt;&lt;!--Axel Kannenberg (2019) ECJ: German ancillary copyright law invalid for publishers, heise online--&gt;{{cite Q|Q124051681|title= ECJ: German ancillary copyright law invalid for publishers}}&lt;/ref&gt; Before that ECJ decision, Google had removed newspapers from Google News in Germany. German publishers then reached an agreement with Google after traffic to their websites plummeted.&lt;ref&gt;&lt;!--Dominic Rushe (2014) Google News Spain to close in response to story links 'tax', Guardian--&gt;{{cite Q|Q124051847}}&lt;/ref&gt; Building on that and similar experience in Spain, the European Union adopted a [[w:Directive on Copyright in the Digital Single Market|Directive on Copyright in the Digital Single Market]] in 2019. A similar link tax proposal in Canada led [[w:Meta|Meta]], the parent company of Facebook, to withdraw news from Canada, and Google agreed to 'pay about $100 million a year into a new fund to support “news&quot;' in Canada. As of 2023-11-30, California was still considering a link tax.&lt;ref&gt;&lt;!--Ken Doctor (2023) Forget the link tax. Focus on one key metric to “save local news, NiemanLab--&gt;{{cite Q|Q124051930}}&lt;/ref&gt;

== Threats from social media == 

The growth of social media has been wonderful and terrible.  It has been wonderful in making it easier for people to maintain friendships and family ties across distances.&lt;ref&gt;Friedland (2017) noted that the Internet works well at the global level, helping people get information from any place in the world, and at the micro level, e.g., with Facebook helping people with similar diseases find one another.  It does not work well at the '“meso level arenas of communication” in the middle.  They're not big enough to aggregate all the scale that goes into creating a worldwide web or even a Wikipedia. See also Lloyd and Friedland (2016).&lt;/ref&gt; 

But it has also been terrible as  &quot;antisocial media&quot;&lt;ref&gt;Vaidhyanathan (2018).&lt;/ref&gt; have been implicated in the relatively recent rise in dysfunctional and counterproductive political polarization and violence. Ding et al. (2023) document, &quot;Same words, different meanings&quot; in their use by [[w:CNN|CNN]] and [[w:Fox News|Fox News]] and how that has interacted with word usage on Twitter between 2010 and 2020 to increase political polarization, &quot;impeding rather than supporting online democratic discourse.&quot;&lt;ref&gt;See also Ashburn.&lt;/ref&gt; 

Extreme examples of this increase have included violent efforts to prevent peaceful transitions of power in the US&lt;ref&gt;Wikipedia &quot;[[w:January 6 United States Capitol attack|January 6 United States Capitol attack]]&quot;, accessed 2023-05-09.&lt;/ref&gt; and Brazil.&lt;ref&gt;Wikipedia &quot;[[w:2023 Brazilian Congress attack|2023 Brazilian Congress attack]]&quot;, accessed 2023-05-09.&lt;/ref&gt; These changes even threaten the national security of the US and its allies,&lt;ref&gt;McMaster (2020). Zuboff (2019) noted that data on many aspects of ordinary daily life are captured and used by people with power for various purposes.  For example, data on people's locations captured from their mobile phones are used to try to sell them goods and services.  Data on a child playing with a smart Barbie doll are used to inculcate shopping habits in child and caregiver.  If you are late on a car payment, your keys can be deactivated until a tow truck can arrive to haul it away.  To what extent do the major media today have conflicts of interest in honestly reporting on this?  How might the experiments proposed herein impact the commercial calculus of major media and the political economy more generally?&lt;/ref&gt; according to [[w:H. R. McMaster|H. R. McMaster]],&lt;ref&gt;Wikipedia &quot;[[w:H. R. McMaster|H. R. McMaster]]&quot;, accessed 2023-05-09.&lt;/ref&gt; President Trump's second national security advisor. 

Various responses to these concerns have been suggested, beyond the recommendations of McChensey and Nichols.  These include the following:&lt;ref&gt;See also the section on &quot;&quot;[[International Conflict Observatory#Suggested responses to these concerns|Suggested reponses to these concerns]]&quot; in the Wikiversity article on &quot;[[International Conflict Observatory]]&quot;.&lt;/ref&gt;:

* Make internet companies liable for defamation in advertisements, similar to print media and broadcasting.&lt;ref&gt;See Baker (2020) and the Wikiversity article on &quot;[[Dean Baker on unrigging the media and the economy]], accessed 2023-07-26.&lt;/ref&gt;  
* Tax advertising revenue received by large internet companies and use that to fund more local media.&lt;ref&gt;Karr and Aaron (2019).&lt;/ref&gt; 
* Replace advertising as the source of funding for social media with subscriptions.&lt;ref&gt;Frank (2021) wrote, &quot;[D]igital aggregators like Facebook ... make money not by charging for access to content but by displaying it with finely targeted ads based on the specific types of things people have already chosen to view. If the conscious intent were to undermine social and political stability, this business model could hardly be a more effective weapon. ... [P]olicymakers’ traditional hands-off posture is no longer defensible.&quot;&lt;/ref&gt; 

To these suggestions, we add the following:  

* Allow some of but not all citizen-directed subsidies for news to go to social media outlets, as suggested below.  
* Require that all organizations whose income depends on promoting or &quot;boosting&quot; content, whether in advertisements or &quot;underwriting spots&quot; or [[w:clickbait|clickbait]], to provide copies of the ads, underwriting spots and clickbait to a central repository like the [[w:Internet Archive|Internet Archive]].  
* Use advertising to discuss overconfidence and encourage people to talk politics with humility and respect, recognizing that the primary differences they have with others are the media they consume.&lt;ref&gt;For studies of ad campaigns in other contexts, see Piwowarski et al. (2019) and Tom-Yov (2018), cited above in discussing &quot;Reducing political polarization&quot;.&lt;/ref&gt;

== How to counter political polarization ==

More research seems to be needed on how to counter the relatively recent increases in political polarization. For example, might some form of [[w:Fairness doctrine|fairness doctrine]]&lt;ref name=fairness/&gt; help reduce political polarization? [[w:Fairness doctrine#Opposition|Conservative leaders are vehemently opposed]], insisting it would be an attack on First Amendment rights. However, as noted above, the tabloid media of Germany seems to have contributed to Hitler's rise to power between 1924 and 1933. How is the increase in political polarization since 1987 and 2004 different from the disregard for fairness of the news media that helped bring Hitler to power? One example: The lawsuit ''[[w:Dominion Voting Systems v. Fox News Network|Dominion Voting Systems v. Fox News Network]]'' was settled with Fox agreeing to to pay Dominion $787.5 million while acknowledging that Fox had knowingly and intentionally made false and defamatory statements about Dominion to avoid losing audience to media outlets that continued to claim fraudulently that Donald Trump not Joe Biden had won the 2020 US presidential election. The settlement permitted Fox to avoid apologizing publicly, which could have threatened their audience share. That settlment was less than 6 percent of Fox's 2022 revenue of $14 billion.&lt;ref&gt;&lt;!--Fox earnings release for the quarter and fiscal year ended June 30, 2022--&gt;{{cite Q|Q124003735}}&lt;/ref&gt; Evidently, ''if that decision made a difference of 6 percent in their audience ratings, Fox made money from defaming Dominion even after paying them $787.5 million.'' If so, it was a good business decision, especially since they did not have to publicly apologize.  

To what extent did Fox's lies about Dominion contribute to the [[w:January 6 United States Capitol attack|mob attacks on the US Capitol on January 6, 2021]], trying to prevent the US Congress from officially declaring that Joe Biden had won the 2020 elections? And what are elected officials prepared to do to improve understanding of what contributes to increases in political polarization and how political differences can be made less lethal and more productive?

== McChesney and Nichols' Local Journalism Initiative ==

As noted above, McChesney and Nichols (2021, 2022) propose a &quot;Local Journalism Initiative&quot;, distributing 0.15 percent of GDP to local news nonprofits via local elections.  They based this partly on their earlier work suggesting that subsidies for newspapers in the US in 1840 was around 0.2 percent of GDP.&lt;ref&gt;McChesney and Nichols (2010, 2016).&lt;/ref&gt;

=== McChesney and Nichols' eligibility criteria === 

To be eligible, McChesney and Nichols say the recipient of such funds should satisfy the following:&lt;ref&gt;McChesney and Nichols (2021, 2022).  They also suggest having the US Postal Service administer this with elections every three years.&lt;/ref&gt; 
* Be a local nonprofit with at least six months of history, so voters could know their work.  
* Be locally based with at most 75 percent of salaries going to local residents.  
* Be completely independent, not a subsidiary of a larger organization.
* Produce and publish original material at least five days per week on their website for free, explicitly in the public domain. 
* Each voter is asked to vote for at least three different local news outlets to support diversity. 
* No single news outlet should get more than 25 percent of that jurisdiction's annual budget for local news subsidies.  
* Each recipient of these subsidies should get at least 1 percent of the vote to qualify, or 0.5 percent of the vote in political jurisdictions with over 1 million people. Diversity and competition are crucial.  
* There will be no content monitoring:  Government bureaucrats will not be allowed to decide what is &quot;good journalism&quot;.  That's up to the voters.  
* Voting would be limited to those 18 years and older.  

=== Alternatives === 

Some aspects of this might be relaxed for at least some political jurisdictions included in an experiment.  For example, might it be appropriate to allow for-profit news outlets to compete for these subsidies as long as they meet the other criteria?&lt;ref&gt;Kaiser (2021) noted that nonprofits in the US cannot endorse political candidates and are limited in how they can get involved in debates on political issues.  Do restrictions like these contribute to the general welfare?  Or might the public interest be better served with citizen-directed subsidies for media that might be more partisan?  This is one more question that might be answered by appropriate experimentation.&lt;/ref&gt; However, we prefer to retain the rules requiring recipients to be local and completely independent, at least for many experimental jurisdictions.&lt;ref&gt;Various contributors to Islam et al., eds. (2002) raised questions about concentrations of power in large media organizations, especially Herman, ch. 4, pp. 61-81 (73-97/336 in pdf).  Djankov et al. (2002) found that &quot;Government ownership of the media is detrimental to economic, political, and-most strikingly-social outcomes&quot;, including education and health.&lt;/ref&gt; If citizen-directed subsidies for local news go to for-profit organizations, to what extent should their finances be transparent, e.g., otherwise complying with the rules for 501(c)(3)s? Might it also be appropriate to allow some portion of these funds to be distributed to noncommercial ''social media'' outlets that submitted all their content to a public, searchable database like the [[w:Internet Archive|Internet Archive]]?   

News written by people paid with these subsidies should be available under a free license like Creative Commons Attribution-ShareAlike (CC BY-SA) 4.0 International license but not necessarily in the public domain:  Other media outlets should be free to further disseminate the news while giving credit to the organization that produced it.   

Many countries have some form of [[w:community radio|community radio]].  Some of those radio stations include what they call news and / or public affairs, and some of those are made available as podcasts via the Internet.&lt;ref&gt;In the US, many of these stations collaborate via organizations such as the [[w:National Federation of Community Broadcasters|National Federation of Community Broadcasters]], the [[w:List of Pacifica Radio stations and affiliates#Radio Stations#Affiliates|Pacifica Network Affiliates]], and the [[w:Grassroots Radio Coalition|Grassroots Radio Coalition]]. One such station with regular local news produced by volunteers in [[w:KBOO|KBOO]] in [[w:Portland, Oregon|Portland, Oregon]]; see Loving (2019).&lt;/ref&gt; If their &quot;news &amp; public affairs&quot; programs are subsequently posted to a website as podcasts, preferably accompanied by some text if not complete transcripts, under a license no more restrictive than CC BY-SA, that should make them eligible for subsidies under the criteria mentioned above if they add at least one new podcast of that nature five days per week.  If the programming of this nature that they produce is ''not'' available on the web or under an appropriate license, part of any experiments as discussed here might include offers to help such radio stations become eligible.  

Might it be wise to allow children to vote for news organizations they like?  Ryan Sorrell, founder and publisher of the Kansas City Defender, insists that, &quot;young people ... are very interested in news. It just has to be produced and packaged the right way for them to be interested in consuming it&quot;.&lt;ref&gt;Holmes (2022).&lt;/ref&gt; The French-language [[:fr:w:Topo (revue)|''Topo'']] present news and complex issues in comic strip format. Their co-editor in chief insists, &quot;there are plenty of ways to get young people interested in current affairs&quot;.&lt;ref&gt;Biehlmann (2023).&lt;/ref&gt; Might allowing children to vote for news outlets they like increase public interest in learning and in civic engagement among both children and their caregivers?  Should this be tested in some experimental jurisdictions?&lt;ref&gt;We may not want infants who cannot read a simple children's book to vote for &quot;news&quot;, but if they can read the names of eligible local news outlets on a ballot, why not encourage them to vote?  As Rourmeen Islam wrote in 2002, &quot;erring on the side of more freedom rather than less would appear to cause less harm.&quot; (World Bank, 2002, pp. 21-22; 33-34/336 in pdf).&lt;/ref&gt;

Some of the money may go to media outlets that seem wacko to many voters.  However, how different might that be from the current situation? Most importantly, if these subsidies have the effect that Tocqueville reported from 1831, they should be good for democracy and for broadly shared peace and prosperity for the long term:  They could stimulate public debate, and wacko media might have ''less'' power than they currently do, with &quot;each separate journal exercis[ing] but little authority; but the power of the periodical press [being] second only to that of the people.&quot;&lt;ref&gt;Tocqueville (1835; 2001, p. 94).&lt;/ref&gt; 

Tocqueville's comparison of newspapers in France and the US in 1831 is echoed in Cagé's (2022) concern about &quot;the Fox News effect&quot; in the US and that of Bolloré in France.  She cites research claiming that biases in Fox News made major contributions to electing Republicans in the US since 2000.&lt;ref&gt;Cagé (2022, pp. 21-22, 59-60).  She cited DellaVigna and Ethan Kaplan (2007), who reported that Fox News had introduced cable programming into 20 percent of town in the US between 1996 and 2000.  They found that the presence of Fox increased the vote share for Republicans between 0.4 and 0.7 percentage points over neighboring non-Fox towns that seemed otherwise indistinguishable.  In 2000 Fox News was available in roughly 35 percent of households, which suggests that Fox News shifted the nationwide vote tally by between 0.15 and 0.2 percentage points.  They conclude that this shift was small but likely decisive in the close 2000 US presidential election.&lt;/ref&gt; These shifts, including changes by the conservative-leaning broadcasting company, Sinclair Broadcast Group, reportedly made a substantive contribution to the election of [[w:Donald Trump|Donald Trump]] as US President in 2016, while a comparable estimate of the impact of changes in MSNBC &quot;is an imprecise zero.&quot;&lt;ref&gt;Cagé (2022, pp. 21-22).  Miho (2022) analyzes the timing of the introduction of biased programming by the conservative-leaning broadcasting company, Sinclair Broadcast Group, between 1992 and 2020, comparing counties in the US with and without a Sinclair station.  This work estimates a 2.5 percentage point increase in the Republican vote share during the 2012 US presidential election and double that during the 2016 and 2020 presidential elections with comparable increases in Republican representation in the US Congress.&lt;/ref&gt; In France, she provides documentation claiming that the media empire of French billionaire [[w:Vincent Bolloré|Vincent Bolloré]] has made a major contribution to the rise of far-right politician [[w:Éric Zemmour|Éric Zemmour]] and is buying media in Spain.&lt;ref&gt;Cagé (2022, pp. 24, 60)&lt;/ref&gt; The pattern is simple:  Fire journalists and replace them with talk shows, which are cheaper to produce and are popular, evidently exploiting the [[w:overconfidence effect|overconfidence effect]].  

To what extent is the increase in political polarization since 1987&lt;ref name=fairness/&gt; and 2004&lt;ref&gt;Wikipedia &quot;[[w:Facebook|Facebook]]&quot;, accessed 2023-07-21.&lt;/ref&gt; due to increased concentration of ownership of both traditional and social media (and how those organizations make money selling changes in audience behaviors to the people who give them money)? To the extent that this increase in polarization has been driven by those changes in the media, citizen-directed subsidies for diverse news should reverse that trend. This hypothesis can be tested by experiments like those proposed herein.

== Roadmap for local news == 

Green et al. (2023) describe &quot;an emerging approach to meeting civic information needs&quot; in a &quot;Roadmap for local news&quot;. This report insists that society needs &quot;civic information&quot;, not merely &quot;news&quot;. It summarizes interviews with 51 leaders from nonprofit and commercial media across all forms of distribution (print, radio, broadcast, digital, SMS) in member organizations, news networks, news funders and researchers. They say that, &quot;Rampant disinformation is being weaponized by extremists&quot;, and &quot;Democratic participation and representation are under threat.&quot; They recommend four strategies to address &quot;this escalating information crisis&quot;: 

# Coordinate work around the goal of expanding “civic information,” not saving the news business;
# Directly invest in the production of civic information;
# Invest in shared services to sustain new and emerging civic information networks; and
# Cultivate and pass public policies that support the expansion of civic information while maintaining editorial independence. 

Part of the motivation for this article on &quot;Information is a public good&quot; is the belief that solid research on the value of such interventions should both (a) make it easier to get the funding needed, and (b) help direct the funding to interventions that seem to make the maximum contributions to improving broadly shared peace and prosperity for the long term at minimum cost.

== Budgets for experiments == 

What factors should be considered in evaluating budgets for experiments to estimate the impact of citizen-directed subsidies for news?   

[[File:Advertising as a percent of Gross Domestic Product in the United States.svg|thumb|Figure 9.  Advertising as a percent of Gross Domestic Product in the United States, 1919 to 2007.&lt;ref name=ads&gt;Galbi (2008).&lt;/ref&gt;]]  

Rolnik et al. (2019) suggested that $50 per person, roughly 0.08 percent of US GDP, might be enough.  However, that's a pittance compared to the revenue lost by newspapers in the US since 1955, as documented in Figure 8 above.  It's also a pittance compared to the money spent on advertising (see Figure 9):  Can we really expect local media funded with only 0.08 or 0.15 percent of GDP to compete with media funded by 2 percent of GDP?  Maybe, but that's far from obvious.  Might it be prudent to fund local journalism in some experimental jurisdictions at levels exceeding the money spent on advertising, i.e., at roughly 2 percent of GDP or more?  If information is a public good, as suggested by the research summarized here, then such high subsidies would be needed in some experimental jurisdictions, because the maximum of anything (including net benefits = benefits minus costs) cannot be confidently identified without conducting some experiments ''beyond the point of diminishing returns''.&lt;ref&gt;A parabola can be estimated from three distinct points.  However, in fitting a parabola or any other mathematical model to empirical data, one can never know if an empirical phenomenon has been adequately modeled and a maximum adequately estimated without data near the maximum and on both sides of it (unless the maximum is at a boundary, e.g., 0).  See, e.g., Box and Draper (2007).&lt;/ref&gt;  

[[File:AccountantsAuditorsUS.svg|thumb|Figure 10.  Accountants and auditors as a percent of the US workforce.&lt;ref name=actg&gt;Accountants and auditors as a percent of US households, 1850 - 2016, using the OCC1950 occupation codes in a sample of households available from from the [[w:IPUMS|Integrated Public Use Microdata Series at the University of Minnesota (IPUMS)]].  For more detail see the &quot;AccountantsAuditorsPct&quot; data set in the &quot;Ecdat&quot; package and the &quot;AccountantsAuditorsPct&quot; vignette in the &quot;Ecfun&quot; package available from within the [[w:R (programming language)|R (programming language)]] using 'install.packages(c(&quot;Ecdat&quot;, &quot;Ecfun&quot;), repos=&quot;http://R-Forge.R-project.org&quot;)'.&lt;/ref&gt;]] 

Also, news might serve a roughly comparable function to accounting and auditing, as both help reduce losses due to incompetence, malfeasance and fraud.  Two points on this:  

# CONTROL FRAUDS:  Black (2013) noted that many heads of organizations can find accountants and auditors willing to certify accounting reports they know to be fraudulent. Black calls such executives &quot;control frauds.&quot;&lt;ref&gt;Black (2013). &lt;/ref&gt; Primary protections against these kinds of problems are vigorous, independent journalists and more money spent on independent evaluations beyond the control of such executives.  In this regard, we note two major differences between the [[w:Savings and loan crisis|Savings &amp; Loan scandal]] of the late 1980s and early 1990s&lt;ref&gt;Wikipedia &quot;[[w:Savings and loan crisis|Savings and loan crisis]]&quot;, accessed 2023-06-25.&lt;/ref&gt; and the [[w:2007–008 financial crisis|international financial meltdown of 2007-2008]]:&lt;ref&gt;Wikipedia &quot;[[w:2007–008 financial crisis|2007–008 financial crisis]]&quot;, accessed 2023-06-25.&lt;/ref&gt; First the major banks by 2007 were much bigger and controlled much larger advertising budgets than the Saving &amp; Loan industry did 15-20 years earlier.  This meant that major media had a much bigger conflict of interest in honestly reporting on questionable activities of these major accounts.  Second, the major banks had made much larger political campaign contributions to much larger portions of both the US House and Senate. However, might the massive amounts of big money spent on campaign finance have been as effective if the major media did not have such a conflict of interest in exposing more details of the corrosive impact of major campaign donors on the quality of government?  To what extent might this corrosive impact be quantified in experimental polities?  
# ADEQUATE RESEARCH OF OUTCOMES:  Many nonprofits and governmental agencies officially have outcome measures, but many of those measures tend to be relatively superficial like the number of people served.  It's much harder to evaluate the actual benefits to the people served and to society.  For example, the Perry Preschool&lt;ref&gt;Schweinhart et al. (2005).  See also Wikipedia &quot;[[w:HighScope|HighScope]]&quot;, accessed 2023-06-15. &lt;/ref&gt; and Abecedarian&lt;ref&gt;e.g., Sparling and Meunier (2019).  See also Wikipedia &quot;Abecedarian Early Intervention Project&quot;, accessed 2023-06-25.&lt;/ref&gt; programs divided poor children and caregivers into experimental and control groups and followed them for decades to establish that their interventions were enormously effective.&lt;ref&gt;For more recent research on the economic value of high quality programs for early childhood development, see, e.g., &lt;!-- &quot;The Heckman Equation&quot; website (heckmanequation.org)--&gt;{{cite Q|Q121010808}}, accessed 2023-07-29.&lt;/ref&gt; Meanwhile, US President Lyndon Johnson's [[w:Great Society|Great Society]] programs,&lt;ref&gt;Wikipedia &quot;[[w:Great Society|Great Society]]&quot;, accessed 2023-07-11.&lt;/ref&gt; and [[w:Head Start|Head Start]] in particular, did not invest as heavily in research.  That lack of documentation of results made them a relatively easy target for political opponents claiming that government is the problem, not the solution. These counter arguments were popularized by US President Ronald Reagan and UK Prime Minister Margaret Thatcher to justify reducing or eliminating government funds for many such programs. Banerjee and Duflo (2019) summarized relevant research in this area by saying that the programs were not the disaster that Reagan, Thatcher, and others claimed, but they were also not as efficient and effective as they could have been, because many local implementations were underfunded, poorly managed and poorly evaluated.   Bedasso (2021) analyzed World Bank projects completed from 2009 to 2020, concluding that high quality monitoring and evaluation on average made a major contribution to the positive results from the successful projects studied.&lt;ref&gt;See also Raimondo (2016).&lt;/ref&gt; To what extent might citizen-directed subsidies for local media as suggested here improve the demand for (and the supply of) better evaluations, leading to better programs and the lower crime, etc., that came from those programs?  To what extent might this effect be quantified using randomized controlled trials comparing different jurisdictions, analogous to the research for which Banerjee, Duflo, and Kramer won the 2019 Nobel Memorial Prize in Economics?  

This discussion makes us wonder if better research and better news might deliver dramatically more benefits than costs in reducing money wasted on both funding wasteful programs and on failing to fund effective ones?  In particular, might society benefit from matching the 1 percent of the workforce occupied by accountants and auditors with better research and citizen-directed subsidies for news (see Figure 10)?  If, for example, 1 or 2 percent of GDP distributed to local news nonprofits via local elections, as described above, increased the average rate of economic growth in GDP per capita by 0.1 percentage point per year, that increase would accumulate over time, so that after 10 or 20 years, the news would in effect become free, paid by money that implementing political jurisdictions would not have without those subsidies. Moreover those accumulations might remain as long as they were not wiped out by events comparable to the economic disasters documented above in discussing &quot;Stalin and Putin&quot; -- and maybe not even then as suggested by Figure 7. 

=== Other recommendations and natural experiments === 

Table 1 compares the recommendations of McChesney and Nichols (2021, 2022) and Rolnik et al. (2019) with other possible points of reference. Crudely similar to McChesney, Nichols, and Rolnik et al., Karr (2019) and Karr and Aaron (2019) recommend &quot;a 2 percent ad tax on all online enterprises that in 2018 earned more than $200 million in annual digital-ad revenues&quot;. They claim that this &quot;would yield more than $1.8 billion a year&quot;,  which is very roughly 0.008 percent of GDP, $5 per person per year;&lt;ref name=Karr&gt;Karr (2019), Karr and Aaron (2019). US GDP for 2019 was $21,381 billion, per International Monetary Fund (2023). Thus, $1.8 billion is 0.0084% of US GDP and $5.44 for each of the 330,513,000 humans in the US in 2019; round to 0.008% and $5 per capita.&lt;/ref&gt; Google has negotiated agreements similar to this with the governments of Australia and Canada.&lt;ref&gt;Hermida (2023).&lt;/ref&gt; Other points of reference include the percent of GDP devoted to accounting and auditing and advertising. As displayed in Figure 10, accountants and auditors are roughly 1 percent of the workforce in the US. It's not clear how to translate that into a percent of GDP, but 2 percent seems like a reasonable approximation, if we assume that average income of accountants and auditors is a little above the national norm and overhead is not quite double their salaries; this may be conservative, because many accountants and auditors have support staff, who are not accountants nor auditors but support their work. 

Another point of reference is the average annual growth rate in GDP per capita since World War II: A subsidy of 2 percent of GDP would be roughly one year's increase in average annual income since World War II, as noted with Figure 1 above. More precisely, the US economy (GDP per capita adjusted for inflation) was 2.2 percent per year between between 1950 and 1990 but only 1.1 percent between 1990 and 2022, according to inequality expert [[w:Thomas Piketty|Thomas Piketty]], who attributed that slowing in the rate of economic growth to the increase in income inequality in the US since 1975, documented in Fgure 6 above. Whether Piketty is correct or not, if 2 percent per year subsidies for journalism close the gap between 1.1 and 2.2 percent per year, those media subsidies would effectively become free after two years, paid out of income the US would not have without them. This reinforces the main point of this essay regarding the need for randomized controlled trials on any intervention with a credible claim to improving the prospects for broadly shared economic growth for the long term.&lt;ref name=GDP&gt;The growth in US GDP per capita is discussed in the working paper on Wikiversity titled, &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;, accessed 2021-05-19.   For a similar comment about an intervention that increased the rate of economic growth becoming free, paid out of income we would not have without it, has been made about the impact of improving education by {{cite Q|Q56849246}}&lt;!-- Endangering Prosperity: A Global View of the American School--&gt;, p. 12.&lt;/ref&gt;    
This table includes other interventions for which humanity would benefit from more substantive evaluation of their impact. This includes [[w:Democracy voucher|Seattle's &quot;Democracy Voucher&quot; program]], which gives each registered voter four $25 vouchers, totaling $100, which they could give to eligible candidates running for municipal office. However, only the first 47,000 were honored;  this limited the city's commitment to $4.7 million every other year.&lt;ref name=Berman&gt;Berman (2015). The Wikipedia article on [[w:Seattle]] says that the gross metropolitan product (GMP) for the Seattle-Tacoma metropolitan area was $231 billion in 2010 for a population of 3,979,845.  That makes the GMP per capita roughly $58,000.  However, the population of Seattle proper was only 608,660 in 2010, making the Gross City Product roughly $35 billion.  $4.7 million is 0.0133 percent of $35 billion.  However, that's very other year, so it's really only 0.007 percent of the Gross City Product.&lt;/ref&gt; If Seattle can afford $100 per registered voter, many other governmental entities can afford something very roughly comparable for each adult in their jurisdiction. Seattle's &quot;democracy vouchers&quot; are used to fund political campaigns, not local media; they are mentioned here as a point of comparison. 

Other interventions that seem to deserve more research than we've seen are the [[w:New Jersey Civic Information Consortium|New Jersey Civic Information Consortium]] (NJCIC)&lt;ref name=njcicBudget&gt;Karr (2020). Per [[w:New Jersey]] the Gross State Product in 2018 was roughly $640 billion;  it's population in 2020 was roughly 9.3 million. The initial  $500,000 for the project is only $0.05 per person per year and only 0.00008 percent of $640 billion.&lt;/ref&gt; and a program in California to improve local news in communities in dire need of strong local journalism. The NJCIC was initially funded at $500,000, which is only 0.00008 percent of New Jersey economy (GDP) of $640 billion. In 2022, the state of California authorized $25 million for up to 40 Berkeley local news fellowships offering &quot;a $50,000 annual stipend [for 3 years] to supplement their salaries while they work in California newsrooms covering communities in dire need of strong local journalism.&quot; This Berkeley program is roughly $0.21 per person per year, 0.0007 percent of the Gross State Produce of $3.6 trillion that year, for an annual rate of very roughly 0.0002 percent of the Gross State Product.&lt;ref name=Berkeley&gt;Natividad (2023) discusses the Berkeley local news fellowships. California Gross State Product from US Bureau of Economic Analysis (2023). California population on 2022-07-01 from US Census Bureau (2023).&lt;/ref&gt; A similar project in Indiana funded by philanthropies began as the Indiana Local News Initiative&lt;ref&gt;Greenwell (2023).&lt;/ref&gt; and has morphed into Free Press Indiana.&lt;ref&gt;See &quot;[https://www.localnewsforindiana.org LocalNewsForIndiana.org]&quot;; accessed 29 December 2023.&lt;/ref&gt;

Some local [[w:League of Women Voters|Leagues of Women Voters]] have all-volunteer teams who observe official meetings of local governmental bodies and write reports.&lt;ref&gt;Wilson (2007).&lt;/ref&gt; The [[w:City Bureau|City Bureau]] nonprofit news organization in Chicago, Illinous, &quot;trains and pays community members to attend local government meetings and report back on them.&quot;&lt;ref&gt;See &quot;[https://www.citybureau.org/documenters-about citybureau.org/documenters-about]&quot;.&lt;/ref&gt; The program has been so successful, it has exanded to other cities.&lt;ref&gt;Greenwell (2023).&lt;/ref&gt;

For an international comparison, we include [[w:amaBhungane|amaBhungane]],&lt;ref name=amaBhu&gt;The budget for [[w:AmaBhungane#Budget|amaBhungane]] in 2020 was estimated at 590,000 US dollars at the current exchange rate, per analysis in the [[w:AmaBhungane#Budget|budget]] section of the Wikipedia article on amaBhungane. That's 0.00017 percent of South African's nominal GDP for that year of 337.5 million US dollars, per the section on &quot;[[w:Economy of South Africa#Historical statistics 1980–2022|Historical statistics 1980–2022]]&quot; in the Wikipedia article on [[w:Economy of South Africa|Economy of South Africa]]; round that to 0.002 percent for convenience. The population of South Africa that year was estimated at 59,309,000, according to the section on &quot;[[w:Demographics of South Africa#UN Age and population estimates: 1950 to 2030|UN Age and population estimates: 1950 to 2030]]&quot; in the Wikipedia article on [[w:Demographics of South Africa|Demographics of South Africa]]; this gives a budget of 1 penny US per capita. (All these Wikipedia articles were accessed 2023-12-28.)&lt;/ref&gt; whose investigative journalism exposed a corruption scandal that helped force South African President [[w:Jacob Zuma|Jacob Zuma]] to resign in 2018; amaBhungane's budget is very roughly one penny US per person per year in South Africa, 0.0002 percent of GDP. To the extent that this essay provides a fair and balanced account of the impact of journalism on political economy, South African and the rest of the world would benefit from more funding for amaBhungane and other comparable investigative journalism organizations. This could initially include randomized controlled trials involving citizen-directed subsidies for local news outlets in poor communities in South Africa and elsewhere, as we discuss further in the rest of this essay. Without such experiments, we are asking for funds based more on faith than science. 

{| class=&quot;wikitable sortable&quot; 
!option / reference
!% of GDP 
!colspan=2|US$
!per …
!
|-
|style=&quot;text-align:left;&quot;|US postal subsidies for newspapers 1840-44 
| 0.21%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $140
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
| &lt;ref name=McC-N2010/&gt;
|-
|style=&quot;text-align:left;&quot;|McChesney &amp; Nichols (2021, 2022)
| 0.15%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $100
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
| &lt;ref name=McC-N2021/&gt;
|-
|[[Confirmation bias and conflict#Relevant research|Rolnik et al.]] 
| 0.08%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $50
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|adult &amp; year
| &lt;ref name=Rolnik/&gt; 
|-
|[[w:Free Press (organization)|Free Press]]
| 0.008%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $6
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref name=Karr/&gt;
|-
|[[w:Democracy vouchers|Democracy vouchers]]
| 0.007% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $100
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|voter &amp; municipal election for the first 47,000
|&lt;ref name=Berman/&gt;
|-
|[[Confirmation bias and conflict#Advertising and accounting|advertising]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref name=ads/&gt;
|-
|[[Confirmation bias and conflict#Advertising and accounting|accounting]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref&gt;As noted with Figure 10 and the discussion above, accountants and auditors are roughly 1 percent of the US workforce, and it seems reasonable to guess that their pay combined with support staff and overhead would likely make them roughly double that, 2 percent, as a portion of GDP.&lt;/ref&gt;
|-
|[[US Gross Domestic Product (GDP) per capita|US productivity improvements]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year (GDP per capita)
|&lt;ref&gt;For an analysis of the rate of growth in US GDP per capita, see the working paper on Wikiversity titled, &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;&lt;/ref&gt;
|-
|[[w:New Jersey Civic Information Consortium|New Jersey Civic Information Consortium]]
| 0.00008%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .05
|person &amp; year
|&lt;ref name=njcicBudget/&gt;
|-
| Berkeley local news fellowships  
| 0.0002%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .21
|person &amp; year
|&lt;ref name=Berkeley/&gt;
|-
|[[w:amaBhungane|amaBhungane]]
| 0.0002%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .01
|person &amp; year in South Africa
|&lt;ref name=amaBhu/&gt;
|}
Table 1. Media subsidies and other points of reference. At the low end, political corruption exposed in part by amaBhungane forced the resignation in 2018 of South African President Zuma on a budget that's very roughly one penny US per person per year. If much higher subsidies of 1 percent of GDP restored an annual growth rate of 2.2 percent per year from the more recent 1.1 percent discussed with Figure 1 above, those subsidies would pay for themselves from one year's growth that the US would not otherwise have.

== Other factors == 

We feel a need here to suggest other issues to consider in designing experiments to improve the political economy:  education, empowering women, free speech, free press, peaceful assembly, and reducing political polarization.

EDUCATION:  Modern research suggests that society might have lower crime&lt;ref&gt;Wang et al. (2022).&lt;/ref&gt; and faster rates of economic growth with better funding for and better research&lt;ref&gt;Hanushek and Woessmann (2015).&lt;/ref&gt; on quality child care from pregnancy through age 17. 

EMPOWERING WOMEN:  Might the best known way to limit and reverse population growth be to empower women and girls?  Without that, might the human population continue to grow until some major disaster reduces that population dramatically?&lt;ref&gt;Roser (2017).&lt;/ref&gt;

FREE SPEECH, FREE PRESS, PEACEFUL ASSEMBLY:  Verbitsky said, &quot;Journalism is disseminating information that someone does not want known; the rest is propaganda.&quot;&lt;ref&gt;Verbitsky (2006, p. 16), author's translation from Spanish.&lt;/ref&gt; This discussion of threats, arrests, kidnappings, and murders of journalists&lt;ref&gt;Monitored especially by the [[w:Committee to Protect Journalists|Committee to Protect Journalists]], as discussed in the Wikipedia article on them, accessed 2023-07-04.&lt;/ref&gt; and violent suppression of peaceful assemblies&lt;ref&gt;Monitored by Freedom House and others.  See, e.g., the Wikipedia article on &quot;[[w:List of freedom indices|List of freedom indices]]&quot;, accessed 2023-07-04.&lt;/ref&gt; encourages us to consider the potential utility of efforts to improve local news, as noted by contributors to Islam et al., eds. (2002), cited above.  Data on such problems should be considered in selecting sites for experiments with citizen-directed subsidies for journalism and in analyzing the results from such experiments. Such data should include the incidence of legal proceedings against journalists and publishers&lt;ref&gt;Including the risks of [[w:trategic lawsuit against public participation|strategic lawsuits against public participation]] (SLAPPs) and other questionable uses of the courts including some documented in the &quot;[[w:Freedom of the Press Foundation#U.S. Press Freedom Tracker|U.S. Press Freedom Tracker]]&quot;, mentioned above.&lt;/ref&gt; as well as threats, murders, etc., in jurisdictions comparable to experimental jurisdictions.  

REDUCING POLITICAL POLARIZATION:  What interventions might be tested that would attempt to reduce political polarization while also experimenting with increasing funding for news through small, diverse news organizations? For example, might an ad campaign feature someone saying, &quot;We don't talk politics&quot;, with a  reply, &quot;We have to talk politics with humility and mutual respect, because the alternative is killing people over misunderstandings&quot;? Might another ad say, &quot;Don't get angry: Get curious&quot;?  What can be done to encourage people to get curious rather than angry when they hear something that contradicts their preconceptions?  How can people be encouraged to talk politics with humility and respect for others, understanding that everyone can be misinformed and others might have useful information?&lt;ref&gt;Wikiversity &quot;[[How can we know?]]&quot;, accessed 2023-07-22, reviews relevant research relating to political polarization.  Yom-Tov et al. (2018) described a randomized-controlled trial that compared the effectiveness of different advertisements &quot;to improve food choices and integrate exercise into daily activities of internet users.&quot; They found &quot;powerful ways to measure and improve the effectiveness of online public health interventions&quot; and showed &quot;that corporations that use these sophisticated tools to promote unhealthy products can potentially be outbid and outmaneuvered.&quot; Similar research might attempt to promote strategies for countering political polarization.  See also Piwowarski  et al. (2019).&lt;/ref&gt;  

FOCUS ON POLITICIANS: Mansuri et al. (2023) randomly assigned presidents of village governments in the state of [[w:Tamil Nadu|Tamil Nadu]] in India to one of three groups with (1) a financial incentive or (2) a certificate with an information campaign (without a financial incentive) for better government or (3) a control group. They found that the public benefitted from both the financial and non-financial incentives, and the non-financial incentives were more cost effective. Might it make sense in some experimental jurisdictions to structure the subsidies for local news by asking voters to allocate, e.g., half their votes for local news to outlet(s) that they think provide the best information about politicians with the other half based on &quot;general news&quot;?&lt;ref&gt;Mansuri et al. (2023).&lt;/ref&gt;

PIGGYBACK ON COMMUNITY AND LOCAL DEVELOPMENT PROGRAMS:  The World Bank (2023) notes that, &quot;Experience has shown that when given clear and transparent rules, access to information, and appropriate technical and financial support, communities can effectively organize to identify community priorities and address local development challenges by working in partnership with local governments and other institutions to build small-scale infrastructure, deliver basic services and enhance livelihoods. The World Bank recognizes that CLD [Community and Local Development]  approaches and actions are important elements of an effective poverty-reduction and sustainable development strategy.&quot; This suggests that experiments in citizen-directed subsidies for news might best be implemented as adjuncts to other CLD projects to improve &quot;access to information&quot; needed for the success of that and follow-on projects.  Such news subsidies should complement and reinforce the quality of monitoring and evaluation, which was &quot;significantly and positively associated with project outcome as institutionally measured at the World Bank&quot;.&lt;ref&gt;Raimondo (2016). See also Bedasso (2021).&lt;/ref&gt;

Simple experiments of the type suggested here might add citizen-directed subsidies for local news to a random selection of Community and Local Development (CLD) projects.  Such experimental jurisdictions should be small enough that the budget for the proposed citizen-directed subsidies for news would be seen as feasible but large enough so appropriate data could be obtained and compared with control jurisdictions not receiving such subsidies for local news.

However, some potential recipients of CLD funding may be in news deserts or with &quot;ghost newspapers&quot;, as mentioned above.  Some may not have at least three local news outlets that have been publishing something they call news each workday for at least six months, as required for the local elections recommended by McChesney and Nichols (2021, 2022), outlined above.  In such jurisdictions, the local consultations that identify community priorities for CLD funding should also include discussions of how to grow competitive local news outlets to help the community maximize the benefits they get from the project.  

The need for &quot;at least three local news outlets&quot; is reinforced by the possibilities that two or three local news outlets may be an [[w:oligopoly|oligopoly]], acting like a monopoly.  This risk may be minimized by working to ''limit'' barriers to entry and to encourage different news outlets to serve different segments of the market for news.  The risks of oligopolistic behavior may be further reduced by requiring all recipients of citizen-directed subsidies to release their content under a free license like the Creative Commons Attribution-ShareAlike (SS BY-SA) 4.0 international license.  This could push each independent local news outlet to spend part of their time reading each other's work while pursuing their own journalistic investigations, hoping for scoops that could attract a wider audience after being cited by other outlets.&lt;ref&gt;Wikipedia &quot;[[w:Oligopoly|Oligopoly]]&quot;, accessed 2023-07-06.&lt;/ref&gt;

This preference for at least three independent local news outlets in an experimental jurisdiction puts a lower bound on the size of jurisdictions to be included as experimental units, especially if we assume that the independent outlets should employ on average at least two journalists, giving a minimum of six journalists employed by local news outlets in an experimental jurisdiction.  The discussions above suggested subsidies ranging from 0.08 percent to 2 percent or more.  To get a lower bound for the size of experimental jurisdictions, we divide 6 by 0.08 and 2 percent:  Six journalists would be 0.08 percent of a population of 7,500 and 2 percent of a population of 300.

== Sampling units / experimental polities == 

Many local governments could fund local news nonprofits at 0.15% of GDP, because it would likely be comparable to what they currently spend on accounting, media and public relations.&lt;ref&gt;&quot;State and local governments [in the US] spent $3.5 trillion on direct general government expenditures in fiscal year 2020&quot;, with states spending $1.7 trillion and local governments $1.8 trillion&quot;, per Urban Institute (2024). The nominal GDP of the US for 2020 was $21 trillion, per International Monetary Fund (2024). Thus, local government spending $1.7 trillion is 1.8% of the $21 trillion US GDP, which is comparable to the money spent on accounting per Figure 10 and advertising per Figure 9.&lt;/ref&gt; If the results of such funding are even a modest percent of the benefits claimed in the documents cited above, any jurisdiction that does that would likely obtain a handsome return on that investment. 

Jurisdictions for randomized controlled trials might include some of the members of the United Nations with the smallest Gross Domestic Products (GDPs) or even some of the poorest census-designated places&lt;ref&gt;Wikipedia &quot;[[w:Census-designated place|Census-designated place]]&quot;, accessed 2023-07-11.&lt;/ref&gt; in a country like the US. Alternatively, they might include areas with seemingly intractable cycles of violence like Israel and Palestine: The budget for interventions like those proposed herein are a fraction of what is being spent on defense and on violence challenging existing power structures. If interventions roughly comparable to those discussed herein can reduce the lethality of a conflict at a modest cost, it would have an incredible return on investment (ROI). That would be true not merely for the focus of the intervention but for other similar conflicts. 

For illustration purposes only, Table 2 lists the six countries in the United Nations with the smallest GDPs in 2021 in US dollars at current prices according to the United Nations Statistics Division plus Palestine and Israel, along with their populations and GDP per capita plus the money required to fund citizen-directed subsidies at 0.15 percent of GDP, as recommended by McChesney and Nichols (2021, 2022).  The rough budgets suggested here would only be for a news subsidy companion to Community and Local Development (CLD) projects, as discussed above or for intervention(s) attempting to reduce the lethality of conflict.  Other factors should be considered in detailed planning.  For example, the budget for such a project in [[w:Montserrat|Montserrat]] may need to be increased to support greater diversity in the local news outlets actually subsidized.  And a careful study of local culture in [[w:Kiribati|Kiribati]] may indicate that the suggested budget figure there may support substantially fewer than the 97 journalists suggested by the naive computations in this table.  The key point, however, is that subsidies of this magnitude would be modest as a proportion of many other projects funded by agencies like the World Bank or the money spent on defense or war. 

{| class=&quot;wikitable sortable&quot; style=text-align:right
! Country !! Population !! GDP / capita
! GDP (million USD) 
! annual subsidy at 0.15% of GDP ($K) 
! number of journalists&lt;sup&gt;(*)&lt;/sup&gt;
|-
| [[w:Tuvalu|Tuvalu]] || 11,204 || $5,370 || $60 || $90 ||8.4
|-
| [[w:Montserrat|Montserrat]] || 4,417 || $16,199 || $72 || $107 || 3.3
|- 
| [[w:Nauru|Nauru]] || 12,511 || $12,390 || $155 || $233 || 9.4
|- 
| [[w:Palau|Palau]] || 18,024 || $12,084 || $218 || $327 || 13.5
|- 
| [[w:Kiribati|Kiribati]] || 128,874 || $1,765 || $227 || $341 ||96.7
|- 
| [[w:Marshall Islands|Marshall Islands]] || 42,040 || $6,111 || $257 || $385 || 31.5
|- 
| [[w:State of Palestine|State of Palestine]] || 5,483,450 || $3,302 || $18,037 || $27,055 || 4,113
|- 
| [[w:Israel| Israel]] || 9,877,280 || $48,757 || $481,591 || $722,387 || 7,408 
|}
Table 2.  Rough estimate of the budget for subsidies at 0.15 percent of GDP for the 6 smallest members of the UN plus Palestine and Israel. Population and GDP at current prices per United Nations Statistics Division (2023). (*) &quot;Number of journalists&quot; was computed assuming each journalist would cost twice the GDP / capita.  For example, the GDP / capita for Tuvalu in this table is $5,370. Double that to get $10,740.  Divide that into $90,000 to get 8.4.  

Other possibilities for experimental units might be historically impoverished subnational groups like [[w:Native Americans in the United States|Native American jurisdictions in the United States]].  As of 12 January 2023 there were &quot;574 Tribal entities recognized by and eligible for funding and services from the [[w:Bureau of Indian Affairs|Bureau of Indian Affairs]] (BIA)&quot;, some of which have multiple subunits, e.g., populations in different counties or census-designated places. For example, the largest is the [[w:Navajo Nation|Navajo Nation Reservation]] that is split between Arizona, New Mexico, and Utah.&lt;ref&gt;Newland (2023).&lt;/ref&gt; Some of these subdivisions are too small to be suitable for experiments in citizen-directed subsidies for news.  Others have subdivisions large enough so that some subdivisions might be in experimental group(s) with others as controls.  If there are at least three subdivisions with sufficient populations, at least one could be a control, with others being Community and Local Development (CLD) projects both with and without companion news subsidies, as discussed above.&lt;ref&gt;Data analysis might consider spatial autocorrelation, as used by Mohammadi et al. (2022) and multi-level time series text analysis, used by Friedland et al. (2022).  The latter discuss &quot;Asymmetric communication ecologies and the erosion of civil society in Wisconsin&quot;:  That state had historically been moderate &quot;with a strong progressive legacy&quot;.  Then in 2010 they elected a governor who attacked the state's public sector unions with substantial success and voted for Donald Trump for President in 2016 but ''against'' him in 2020.&lt;/ref&gt;

== Supplement not replace other funding == 

The subsidies proposed here should supplement not replace other funding, similar to the subsidies under the US Postal Service Act of 1792.  McChesney and Nichols recommended that an organization should be publishing something they call news five days per week for at least six months, so the voters would know what they are voting for.  Those criteria might be modified, at least in some experimental jurisdictions, especially in news deserts, as something else is done to create local news organizations eligible to receive a portion of the experimental citizen-directed subsidies.  

The [[w:Institute for Nonprofit News|Institute for Nonprofit News]] and Local Independent Online News (LION) Publishers&lt;ref&gt;&lt;!-- Local Independent Online News (LION) Publishers--&gt;{{cite Q|Q104172660}}&lt;/ref&gt; help local news organizations get started and maintain themselves.  Organizations like them might help new local news initiatives in experimental jurisdictions as discussed in this article.

== Funding research in the value of local news ==

We would expect two sources of funding for research to quantify the value of local news 

* The [[w:World Bank|World Bank]] has already discussed the value of news. We would expect that organizations that fund community and local development projects would also want to fund experiments in anything that seemed likely to increase the return on their investments in such projects. 
* Folkenflik (2023) wrote, &quot;Some of the biggest names in American philanthropy have joined forces to spend at least $500 million over five years to revitalize the coverage of local news in places where it has waned.&quot; This group of philanthropic organizations includes the American Journalism Project, which says they &quot;measure the impact of our philanthropic investments and venture support by evaluating our efficacy in catalyzing grantees’ organizational growth, sustainability and impact.&quot;&lt;ref&gt;Website of the American Journalism Project accessed 29 December 2023 ([https://www.theajp.org/about/impact/# https://www.theajp.org/about/impact/#]).&lt;/ref&gt; 

If the claims made above for the value of news are fair, then appropriate experiments might be able to quantify who benefit from improving the news ecology, how much they benefit, which structures seem to work the best, and even the optimal level of funding.

== Summary == 

This article has summarized numerous claims regarding different ways in which information is a public good.  Many such claims can be tested in experiments crudely similar to those for which Banerjee, Duflo, and Kremer won the [[w:2019 Nobel Memorial Prize in Economic Sciences|2019 Nobel Memorial Prize in Economics]].  We suggest funding such projects as companions to Community and Local Development (CLD) projects.  If the research cited above is replicable, the returns on such investments could be huge, delivering benefits to the end of human civilization, similar to those claimed for newspapers published in the US in the early nineteenth century, which may have made major contributions to carrying the US to its current position of world leadership and to developing technologies that benefit the vast majority of humanity the world over today.

== Acknowledgements == 

Thanks especially to Bruce Preville who pushed for evidence supporting wide ranging claims of media influence in limiting progress against many societal ills.  Thanks also to Dave Black for suggesting experimenting with Native American jurisdictions in the US and to Joy Ellsworth for describing the substantial cultural challenges that such interventions might face.

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* &lt;!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) --&gt;{{cite Q|Q109978337}}
* &lt;!-- H. R. McMaster (2020) Battlegrounds: The Fight to Defend the Free World (HarperCollins) --&gt;{{cite Q|Q104774898}}
* &lt;!-- Antonela Miho (2020) Small Screen, Big Echo? Estimating the Political Persuasion of Local Television News Bias using Sinclair Broadcast Group as a Natural Experiment--&gt;{{cite Q|Q120895467}}
* &lt;!-- Hosein Mohammadi, Samira Shayanmehr and Juan D. Borrero (2023) &quot;Does Freedom Matter for Sustainable Economic Development? New Evidence from Spatial Econometric Analysis&quot;, Mathematics, 11 --&gt;{{cite Q|Q120736319}}
* &lt;!-- US National Academies of Sciences, Engineering, and Medicine (2023) Nobel Prize Summit Fuels Initiatives to Combat Misinformation and Disinformation and Build Trust in Science--&gt;{{cite Q|Q124711722|author=National Academies}}
* &lt;!-- Ivan Natividad (2022) State funds Berkeley Journalism $25 million to strengthen California's local news coverage--&gt;{{cite Q|Q123537590}}
* &lt;!-- Bryan Newland (2023) &quot;Indian Entities Recognized by and Eligible To Receive Services From the United States Bureau of Indian Affairs&quot;, Bureau of Indian Affairs, US Department of the Interior --&gt;{{cite Q|Q120512083}}
* &lt;!-- Jeremy Nguyen, Abbas Valadkhani, Alan Nguyen, Alexandra Wake (2021) Press Freedom and the Global Economy: The Cost of Slipping Backwards--&gt;{{cite Q|Q121011930}}
* &lt;!-- The Nobel Prize (2019) &quot;Prize in economic sciences 2019&quot; --&gt;{{cite Q|Q119948228|author=Nobel Prize}}
* &lt;!--  Ned Parker, Isabel Coles, Raheem Salma (2014) &quot;Special Report: How Mosul fell - An Iraqi general disputes Baghdad's&quot;, Reuters--&gt;{{cite Q|Q124326278}}
* &lt;!-- M. Piwowarski, U. Shankar Singh, &amp; K. Nermend (2019). &quot;The cognitive neuroscience methods in the analysis of the impact of advertisements in shaping people's health habits&quot; European Research Studies Journal, 22(4), 457-471 --&gt;{{cite Q|Q118087826}}
* &lt;!-- Thomas Piketty (2022) A brief history of equality (Harvard U. Pr.) --&gt;{{cite Q|Q115434513}}
* &lt;!-- Scott Plous (1993) The psychology of judgment and decision making (McGraw-Hill) --&gt;{{cite Q|Q118105219}}
* &lt;!--Kyle Pope (2023-04-17) &quot;Free Press, Functioning Democracy: You can’t have one without the other&quot;, Columbia Journalism Review--&gt;{{cite Q|Q124047437}}
* &lt;!-- Gary W. Potter and Victor E. Kappeler (1998) Constructing Crime: Perspectives on Making News and Social Problems (Waveland Pr.) --&gt;{{cite Q|Q96343487|author=Gary W. Potter and Victor E. Kapeller, eds.}}
* &lt;!-- Estelle Raimondo (2016) &quot;What Difference Does Good Monitoring and Evaluation Make to World Bank Project Performance?&quot;, Policy Research Working Paper 7726, World Bank --&gt;{{cite Q|Q120750870}}
* &lt;!-- Reuters (2023-10-25) Israel to amend budget, Gaza war direct cost at $246 mln daily--&gt;{{cite Q|Q124336510|author=Reuters}}
* &lt;!-- Guy Rolnik; Julia Cagé; Joshua Gans; Ellen P. Goodman; Brian G. Knight; Andrea Prat; Anya Schiffrin (1 July 2019), &quot;Protecting Journalism in the Age of Digital Platforms&quot; (PDF), Booth School of Business --&gt;{{cite Q|Q106465358}}
* &lt;!-- Max Roser (2017) &quot;Fertility Rate&quot;, Our World in Data --&gt;{{cite Q|Q120333634}}
* &lt;!-- Vincent F. Sacco (1995) &quot;Media Constructions of Crime&quot;, Annals Of The American Academy of Political and Social Science, 539:141-154, Wikidata Q56805896, reprinted as ch. 2 of Potter and Kappeler (1998) --&gt;{{cite Q|Q106878177}}, reprinted as ch. 2 in Potter and Capeller (1998).
* &lt;!-- Vincent F. Sacco (2005) When Crime Waves (SAGE) --&gt;{{cite Q|Q96344789}}
* &lt;!-- Fabian Scheidler (2024-03a) Espoirs et misère de la critique des médias en Allemagne, Le Monde diplomatique--&gt;{{cite Q|Q125430621|date=2024a}}
* &lt;!-- Fabian Scheidler (2024-03b) &quot;La disparition d'une école contestataire&quot;, Le Monde diplomatique--&gt;{{cite Q|Q125455715|date=2024b}}
* &lt;!-- Sam Schulhofer-Wohl and Miguel Garrido (2009) &quot;Do newspapers matter?  Short-run and long-run evidence from the closure of the Cincinnati Post&quot;, NBER Working Paper Series 14817 (http://www.nber.org/papers/w14817, 2023-05-05) --&gt;{{cite Q|Q105879512}}
* &lt;!-- Lawrence J. Schweinhart, Jeanne Montie, Zongping Xiang, W. Steven Barnett, Clive R. Belfield, and Milagros Nores (2005) &quot;Lifetime effects: The High/Scope Perry Preschool Study through age 40: Summary, Conclusions, and Frequently Asked Questions&quot;, HighScope --&gt;{{cite Q|Q119527802}}
* &lt;!--Avi Shlaim (2014) The Iron Wall: Israel and the Arab World, 2nd ed. (Penguin)--&gt;{{cite Q|Q124213243}}
* &lt;!-- Joseph Sparling and Kimberly Meunier (24 July 2019). &quot;Abecedarian: An Early Childhood Education Approach that has a Rich History and a Vibrant Present&quot;. International Journal of Early Childhood. 51: 207–216. ISSN 0020-7187--&gt;{{cite Q|Q120552586}}
* &lt;!-- Josh Stearns and Christine Schmidt (2022) How We Know Journalism is Good for Democracy--&gt;{{cite Q|Q123522369}}
* &lt;!-- Joseph E. Stiglitz (1999) &quot;Knowledge as a global public good&quot;, chapter in Kaul et al. (1999), pp. 308-315 --&gt;{{cite Q|Q120732906}}
* &lt;!-- Joseph Stiglitz (2002) Transparency in Government--&gt;{{cite Q|Q120941389}}
* &lt;!-- Leigh Thompson (2020) Heart and mind of the negotiator, 7th ed. --&gt;{{cite Q|Q120870057}}
* &lt;!-- Leigh Thompson and George Loewenstein (1992) &quot;Egocentric interpretations of fairness and interpersonal conflict&quot;, Organizational Behavior and Human Decision Processes. 51 (2): 176–197--&gt;{{cite Q|Q29300051}}
* &lt;!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) --&gt;{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}}
* &lt;!-- Rachel Treisman (2021)  When this hedge fund buys local newspapers, democracy suffers--&gt;{{cite Q|Q121438095}}
* &lt;!-- United Nations (2022) World Population Prospects 2022, Online Edition (Department of Economic and Social Affairs, Population Division) --&gt;{{cite Q|Q113015862|author=United Nations|publisher=unset}}
* &lt;!-- United Nations Statistics Division (2021) &quot;Basic Data Selection&quot; --&gt;{{cite Q|Q120485981|author=United Nations Statistics Division|date=2023-07-06|publisher=unset}}
* &lt;!-- US Bureau of Economic Analysis (2023-03-31) Gross Domestic Product by State, 4th Quarter and Annual 2022--&gt;{{cite Q|Q124036982|author=US Bureau of Economic Analysis}}
* &lt;!-- US Census Bureau (2023) Quick facts--&gt;{{cite Q|Q124037268|author=US Census Bureau}}
* &lt;!--US Institute of Peace (2016) Tools for Improving Media Interventions in Conflict Zones--&gt;{{cite Q|Q124692870|author =US Institute of Peace}}
* &lt;!-- Siva Vaidhyanathan (2018) Antisocial Media: How Facebook Disconnects Us and Undermines Democracy (Oxford U. Pr.) --&gt;{{cite Q|Q56027099}}
* &lt;!-- Alexandra Wake, Abbas Valadkhani, Alan Nguyen, and Jeremy Nguyen (2021-04-15) &quot;The Economic Costs of Curbing Press Freedom&quot;, Global Investigative Journalism Network--&gt;{{cite Q|Q120739929}}
* &lt;!-- Leah Wang, Wendy Sawyer, Tiana Herring and Emily Widra (2022) &quot;Beyond the count: A deep dive into state prison populations&quot;, Prison Policy Initiative --&gt;{{cite Q|Q120332620}}
* &lt;!-- Garrett Watson (2022) Combined Federal and State Corporate Income Tax Rates in 2022 --&gt;{{cite Q|Q120793929}}  
* &lt;!-- Richard G. Wilkinson and Kate E. Pickett (2017) &quot;The Enemy between Us: The Psychological and Social Costs Of Inequality&quot;, European Journal of Social Psychology, 47(1)11-24 --&gt;{{cite Q|Q118385024}}
* &lt;!--Mary G. Wilson (2007) Observing Your Government in Action: Protecting your right to know (LWV)--&gt;{{cite Q|Q124048761}}
* &lt;!-- World Bank (2022) Annual Report --&gt;{{cite Q|Q118270948|author=World Bank|series=unset}}
* &lt;!-- World Bank (2023) &quot;Community and Local Development&quot; (web page:  &quot;https://www.worldbank.org/en/topic/communitydrivendevelopment&quot;) --&gt;{{cite Q|Q120380457|author=World Bank|date=2023-03-27|publisher=unset}}
* &lt;!-- Elad Yom-Tov, Jinia Shembekar, Sarah Barclay, and Peter Muennig (2018) &quot;The effectiveness of public health advertisements to promote health: a randomized-controlled trial on 794,000 participants&quot;--&gt;{{cite Q|Q91869157}}
* &lt;!-- Erica York (2023-01-26) &quot;Summary of the Latest Federal Income Tax Data, 2023 Update&quot;, Tax Foundation --&gt;{{cite Q|Q118189145}}
* &lt;!-- Shoshana Zuboff (2019) The Age of Surveillance Capitalism (PublicAffairs) --&gt;{{cite Q|Q75804726}}
* James D. Wolfensohn (2002) Foreword to Islam et al. (2002, pp. v-vi).

== Notes ==
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:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].''  

::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' 

== Abstract == 

This article reviews literature relevant to the claim that &quot;information is a public good&quot; and recommends experiments to quantify the impact of news on society, including on violent conflicts and broadly shared economic growth. We propose randomized controlled trials to evaluate the relative effectiveness of alternative interventions on the lethality of conflict and broadly shared economic growth. Experimental units would be polities in conflict or with incomes (nominal Gross Domestic Products, GDPs or gross local products) small enough so competitive local news outlets could be funded by philanthropies or organizations like the World Bank but large enough that their political economies have been tracked with sufficient accuracy to allow them to be considered in such an experiment. One factor in such experiments would be subsidies for local journalism, perhaps distributed to local news outlets on the basis of local elections, similar to the proposal of McChesney and Nichols (2021, 2022).

== Introduction ==

:''Information is a public good.''&lt;ref&gt;This is the title of Cagé and Huet (2021, in French).  However, the thrust of their book is very different.  It is subtitled, &quot;Refounding media ownership&quot;.  Their focus is on creating legal structure(s) to support journalistic independence as outlined in Cagé (2016).&lt;/ref&gt; 
:''Misinformation is a public nuisance.''&lt;ref&gt;The Wikipedia article on &quot;[[w:Public nuisance|Public nuisance]]&quot; says, &quot;In English criminal law, public nuisance was a common law offence in which the injury, loss, or damage is suffered by the public, in general, rather than an individual, in particular.&quot; (accessed 2023-04-24.)&lt;/ref&gt;
:''Disinformation is a public evil.''&lt;ref&gt;The initial author of this essay is unaware of any previous use of the term, &quot;public evil&quot;, but it seems appropriate in this context to describe content disseminated by mass media, including social media, curated with the explicit intent to convince people to support public policies contrary to the best interests of the audience and the general public.&lt;/ref&gt; 

=== Public goods ===

In economics, a [[w:public good|public good]] is a good (or service) that is both [[w:non-rivalrous|non-rivalrous]] and [[w:non-excludable|non-excludable]].&lt;ref&gt;e.g., Cornes and Sandler (1996).&lt;/ref&gt; Non-rivalrous means that we can all consume it at the same time. An apple is rivalrous, because if I eat an apple, you cannot eat the same apple. 

A printed newspaper may be rivalrous, because it may not be easy for you and me to hold the same sheet of paper and read it at the same time. However, the ''news'' itself is non-rivalrous, because both of us and anyone else can consume the same news at the same time, once it is produced, especially if it's published openly on the Internet or broadcasted on radio or television. 

Non-excludable means that once the good is produced, anyone can use it without paying for it. Information is non-excludable, because everyone can consume it at the same time once it becomes available. [[w:Copyright|Copyright]] law does ''not'' apply to information: It applies to ''expression''.&lt;ref&gt;The US Copyright Act of 1976, Section 102, says, &quot;Copyright protection subsists ... in original works of authorship fixed in any tangible medium of expression ... . In no case does copyright protection ... extend to any idea, procedure, process, system, method of operation, concept, principle, or discovery.&quot; 17 U.S. Code § 102. &lt;!--US Copyright Law of 1976--&gt;{{cite Q|Q3196755}}&lt;/ref&gt; [[w:Joseph Stiglitz|Stiglitz]] (1999) said that [[w:Thomas Jefferson|Thomas Jefferson]] anticipated the modern concept of information as a public good by saying, &quot;He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me.&quot; Stiglitz distinguished between &quot;push and pull mechanisms&quot; to promote innovation and creative work: &quot;Push&quot; mechanisms pay for work upfront, hoping that it will achieve a desired outcome, like citizen-directed subsidies for newspapers. &quot;Pull&quot; mechanisms set a target and then reward those who reach the target, like copyrights and patents.&lt;ref&gt;Baker (2023).&lt;/ref&gt;

Lindahl (1919, 1958) recommended taxing people for public goods in proportion to the benefits they receive. For subsidies for news, especially citizen-directed, this would mean taxing primarily the poor and middle class to fund this.&lt;ref&gt;For more on this, see the Wikipedia articles on [[w:Lindahl tax|Lindahl tax]] and [[w:Theories of taxation|Theories of taxation]].&lt;/ref&gt; If they receive benefits as claimed in the literature cited in this article, the benefits they receive would soon exceed the taxes they pay for it, making the news subsidies effectively free in perpetuity, paid by benefits the poor and middle class would not have without these subsidies. If Piketty (2021, cited below with Figure 1) is correct, the ultra-wealthy would likely also benefit in absolute terms, though the relative distinction between them and the poor would be reduced.

This article recommends [[w:randomized controlled trials|randomized controlled trials]] to quantify the extent to which experimental interventions benefit the public by modifying information environment(s) in ways that (a) reduce political polarization and / or violence and / or (b) improve broadly shared peace and prosperity for the long term.

=== Sharing increases the value === 

The logic behind claiming that &quot;information is a public good&quot; can be easily understood as follows:  

:''If I know the best solution to any major societal problem, it will not help anyone unless a critical mass of some body politic shares that perception.  Conversely, if a critical mass of a body politic believes in the need to implement a certain reform, it will happen, even if I am ignorant of it or completely opposed to it.''  

We can extend this analysis to our worst enemies:  

:It is in ''our best interest'' to help people supporting our worst enemies get information they want, ''independent'' of controls that people with power exercise over nearly all major media today:  If our actions reduce the ability of their leaders to censor their media (and of our political and economic leaders to censor ours), the information everyone gets should make it harder for leaders to convince others to support measures contrary to nearly everyone's best interests.  

What kinds of data can we collect and analyze to evaluate who benefits and who loses from alternative interventions attempting to improve the media? See below.&lt;ref&gt;The power relationship between media and politicians can go both ways.  In addition to asking the extent to which politicians control the media, we can also consider the extent to which political leaders might feel constrained by the major media: To what extent do the major media create the stage upon which politicians read their lines, as claimed in the Wikiversity article on &quot;[[Confirmation bias and conflict]]&quot;?  Might a more diverse media environment make it easier for political leaders to pursue policies informed more by available research and less by propaganda?  Might experiments as described herein help politicians develop more effective governmental policies, because of a reduction in the power of media whose ownership and funding are more diverse? This is discussed further in this article in a section on [[Information is a public good: Designing experiments to improve government#Media and war|Media and war]].&lt;/ref&gt;

=== World Bank on the value of information === 

In 2002 the President of the [[w:World Bank|World Bank]],&lt;ref&gt;The 2022 World Bank Group portfolio was 104 billion USD (World Bank 2022, Table 1, p. 13; 17/116 in PDF).  An improvement of 0.1 percentage points in the performance of that portfolio would be 104 million.  A lot could be accomplished with budgets much smaller than this.&lt;/ref&gt; [[w:James Wolfensohn|James Wolfensohn]], wrote, &quot;[A] free press is not a luxury. It is at the core of equitable development. The media can expose corruption. ... They can facilitate trade [and bring] health and education information to remote villages ... . But ... the independence of the media can be fragile and easily compromised. All too often governments shackle the media. Sometimes control by powerful private interests restricts reporting. ... [T]o support development, media need the right environment{{mdash}}in terms of freedoms, capacities, and checks and balances.&quot;&lt;ref&gt;Wolfensohn (2002). More on this is available in other contributions to Islam et al. (2002) including [[w:Joseph Stiglitz|Stiglitz]] (2002), who noted the following: &quot;There is a natural asymmetry of information between those who govern and those whom they are supposed to serve. ... Free speech and a free press not only make abuses of governmental powers less likely, they also enhance the likelihood that people's basic social needs will be met. ... [S]ecrecy distorts the arena of politics. ... Neither theory nor evidence provides much support for the hypothesis that fuller and timelier disclosure and discussion would have adverse effects. ... The most important check against abuses is a competitive press that reflects a variety of interests. ... [F]or government officials to appropriate the information that they have access to for private gain ... is as much theft as stealing any other public property.&quot;&lt;/ref&gt;

Below please find proposals for evaluating alternative ways of improving the media and circumstances under which they are or are not effective.

=== US Postal Service Act of 1792: a natural experiment === 

[[w:Robert W. McChesney|McChesney]] and [[w:John Nichols (journalist)|Nichols]] (2010, 2016) suggested that the US [[w:Postal Service Act|Postal Service Act]]&lt;ref&gt;Wikipedia &quot;[[w:Postal Service Act|Postal Service Act]]&quot;, accessed 2023-07-11.&lt;/ref&gt; of 1792 made a major contribution to making the US what it is today.  Under that act, newspapers were delivered up to 100 miles for a penny, when first class postage was between 6 and 25 cents depending on distance.  They estimated that between 1840 and 1844, the US postal subsidy was 0.211 percent of GDP with federal printing subsidies adding another 0.005 percent, totaling 0.216 percent of GDP,&lt;ref name=McC-N2010&gt;McChesney and Nichols (2010, pp. 310-311, note 88).&lt;/ref&gt; roughly $140 per person per year in 2019 dollars.&lt;ref name=McN_IMF&gt;International Monetary Fund (2023):  US Gross domestic product per capita at current prices was estimated at $65,077 for 2019 on 2023-04-28. 0.211% of $65,077 = $137; round to $140 for convenience.&lt;/ref&gt; We use 2019 dollars here to make it easy to compare with Rolnik et al. (2019), who recommended $50 per adult per year, which is roughly 0.08 percent of US GDP.  Rolnik et al. added that the level of subsidies would require &quot;extensive deliberation and experimentation&quot;.&lt;ref name=Rolnik&gt;Rolnik et al. (2019, p. 44).  Per [[w:Demographics of the United States]], 24 percent of the US population is under 18, so adults are 76 percent of the population. Thus, $50 per adult is $37.50 per capita. US GDP per capita in $65,0077 in 2019 in current dollars per International Monetary Fund (2023). Thus, $37.50 per capita would be roughly 0.077 percent of GDP; round to 0.08 percent for convenience.&lt;/ref&gt; More recently McChesney and Nichols have recommended 0.15 percent of GDP, considering the fact that the advent of the Internet has nearly eliminated the costs of printing and distribution.&lt;ref name=McC-N2021&gt;McChesney and Nichols (2021; 2022, p. 19).&lt;/ref&gt;

[[w:Alexis de Tocqueville|Tocqueville]], who visited the US in 1831, observed the following:  
* [T]he liberty of the press does not affect political opinion alone, but extends to all the opinions of men, and modifies customs as well as laws. ... I approve of it from a consideration more of the evils it prevents, than of the advantages it insures.&lt;ref&gt;Tocqueville (1835; 2001, p. 91).  In 2002 Roumeen Islam stated this more forcefully:  &quot;Arbitrary actions by government are always to be feared. If there is to be a bias in the quantity of information that is released, then erring on the side of more freedom rather than less would appear to cause less harm.&quot; (World Bank, 2002, pp. 21-22; 33-34/336 in pdf).&lt;/ref&gt;
* The liberty of writing ... is most formidable when it is a novelty; for a people who have never been accustomed to hear state affairs discussed before them, place implicit confidence in the first tribune who presents himself.  The Anglo-Americans have enjoyed this liberty ever since the foundation of the Colonies ... .  A glance at a French and an American newspaper is sufficient to show the difference ... .  In France, the space allotted to commercial advertisements is very limited, and the news-intelligence is not considerable; but the essential part of the journal is the discussion of the politics of the day.  In America, three-quarters of the enormous sheet are filled with advertisements, and the remainder is frequently occupied by political intelligence or trivial anecdotes:  it is only from time to time that one finds a corner devoted to passionate discussions, like those which the journalists of France every day give to their readers.&lt;ref&gt;Tocqueville (1835; 2001, p. 92).&lt;/ref&gt; 
* It has been demonstrated by observation, and discovered by the sure instinct even of the pettiest despots, that the influence of a power is increased in proportion as its direction is centralized.&lt;ref&gt;Tocqueville (1835; 2001, pp. 92-93).&lt;/ref&gt;
* [T]he number of periodical and semi-periodical publications in the United States is almost incredibly large. In America there is scarcely a hamlet which does not have its newspaper.&lt;ref&gt;Tocqueville (1835; 2001, p. 93).&lt;/ref&gt;
* In the United States, each separate journal exercises but little authority; but the power of the periodical press is second only to that of the people ... .&lt;ref&gt;Tocqueville (1835; 2001, p. 94). &lt;/ref&gt;

[[File:Real US GDP per capita in 5 epocs.svg|thumb|Figure 1.  Average annual income (Gross Domestic Product per capita adjusted for inflation) in the US 1790-2021 showing five epochs identified in a &quot;breakpoint&quot; analysis (to 1929, 1933, 1945, 1947, 2021) documented in the Wikiversity article on &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;.&lt;ref&gt;Wikiversity &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;, accessed 2023-07-18.&lt;/ref&gt; Piketty (2021, p. 139) noted, &quot;In the United States, the national income per inhabitant rose at a rate ... of 2.2 percent between 1950 to 1990 when the top tax rate reached on average 72 percent.  The top rate was then cut in half, with the announced objective of boosting growth.  But in fact, growth fell by half, reaching 1.1 percent per annum between 1990 and 2020&quot;.&lt;ref&gt;A more recent review of the literature of the impact of inequality on growth is provided by Jahangir (2023, sec. 3), who notes that some studies have claimed that inequality ''increases'' the rate of economic growth, while other reach the opposite conclusion. However, 'the preponderant academic position is shifting from the argument that “we don’t have enough evidence” and towards seriously addressing and combating economic inequality.'&lt;/ref&gt; Our analysis of US GDP per capita from Measuring Worth do not match Piketty's report exactly, but they are close. We got 2.3 percent annual growth from 1947 to 1990 then 1.8 percent to 2008 and 1.1 percent to 2020. However, we have so far been unable to find a model that suggests that this decline is statistically significant.]]  

To what extent was [[w:Alexis de Tocqueville|Tocqueville's]] &quot;incredibly large&quot; &quot;number of periodical and semi-periodical publications in the United States&quot; due to the US Postal Service Act of 1792? To what extent did that &quot;incredibly large&quot; number of publications encourage literacy, limit political corruption, and help the US of that day remain together and grow both in land area and economically while contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically? To what extent does the enormous power of the US today rest on the economic growth of that period and its impact on the political culture of that day continuing to the present?&lt;ref&gt;Wikiversity &quot;[[The Great American Paradox]]&quot;, accessed 2023-06-12.&lt;/ref&gt; That growth transformed the US into the world leader that it is today;  see Figure 1.  In the process, it generated new technologies that benefit the vast majority of the world's population alive today. If the newspapers Tocqueville read made any substantive contribution to the growth summarized in Figure 1, the information in those newspapers were public goods potentially benefiting the vast majority of humanity ''to the end of human civilization.''&lt;ref&gt;Acemoglu (2023) documents how the power of monopolies and other politically favored groups often distorts the direction of technology development into suboptimal technologies.  Might increasing the funding for more independent news outlets reduce the power of such favored groups and thereby help correct these distortions and deliver &quot;sizable welfare benefits&quot;, e.g., &quot;in the context of industrial automation, health care, and energy&quot;?&lt;/ref&gt;

=== Other economists === 

We cannot prove that the diversity of newspapers in the early US contributed to the economic growth it experienced. Banerjee and Duflo (2019) concluded that no one knows how to create economic growth. They won the 2019 Nobel Memorial Prize in Economics with Michael Kremer for their leadership in using [[w:randomized controlled trials|randomized controlled trials]]&lt;ref&gt;Wikipedia &quot;[[w:Randomized controlled trials|Randomized controlled trials]]&quot;, accessed 2023-07-11.&lt;/ref&gt; to learn how to reduce global poverty.&lt;ref&gt;Wikipedia &quot;[[w:2019 Nobel Memorial Prize in Economic Sciences|2019 Nobel Memorial Prize in Economic Sciences]]&quot;, accessed 2023-06-13.  Nobel Prize (2019).  Amazon.com indicates that distribution of the book started 2019-11-12, twenty-nine days after the Nobel prize announcement 2019-10-14. Evidently the book must have been completed before the announcement.&lt;/ref&gt; More recently, Wake et al. (2021) found evidence that ''the economic costs of curbing press freedom persist long after such freedoms have been restored.''&lt;ref&gt;See also Nguyen et al. (2021).&lt;/ref&gt; And Mohammadi et al. (2022) found that economic growth rates were impacted by civil liberties, economic and press freedom and the economic growth rates of neighbors (spacial autocorrelation) but not democracy. These findings of Mohammidi et al. (2022) and Wake et al. (2021) reinforce Thomas Jefferson's 1787 comment that, &quot;were it left to me to decide whether we should have a government without newspapers, or newspapers without a government, I should not hesitate a moment to prefer the latter.&quot;&lt;ref&gt;From a letter to Colonel Edward Carrington (16 January 1787), cited in Wikiquote, &quot;[[Wikiquote:Thomas Jefferson|Thomas Jefferson]]&quot;, accessed 2023-07-29.&lt;/ref&gt; To what extent might experiments like those recommended in this article either reinforce or refute this claim of Jefferson from 1787?

=== Randomized controlled trials to quantify the value of information === 

This article suggests randomized controlled trials to quantify the impact of citizen-directed subsidies for journalism, roughly following the recommendations of McChesney and Nichols (2021, 2022) to distribute some small percentage of GDP to local news nonprofits ''via local elections''.  Philanthropies could fund such experiments for some of the smallest and poorest places in the world.  Organizations like the World Bank could fund such experiments as adjuncts to a random selection from some list of other interventions they fund, justified for the same reason that they would not consider funding anything without appropriate accounting and auditing of expenditures, as discussed further below.  

Before making suggestions regarding experiments, we review previous research documenting how information might be a public good.

== Previous research ==

Before considering optimal level of subsidies for news, it may be useful to consider the research for which [[w: Daniel Kahneman|Daniel Kahneman won the 2002 Nobel Memorial Prize in Economics]].&lt;ref&gt;Wikipedia &quot;[[w:Daniel Kahneman|Daniel Kahneman]]&quot;, accessed 2023-04-28.&lt;/ref&gt; Most important for present purposes may be that virtually everyone: 

# thinks they know more than they do ([[w:Overconfidence effect|Overconfidence]]),&lt;ref&gt;Wikipedia &quot;[[w:Overconfidence effect|Overconfidence effect]]&quot;, accessed 2023-04-29. Kahneman and co-workers have documented that experts are also subject to overconfidence and in some cases may be worse.  Kahneman and Klein (2009) found that expert intuition can be learned from frequent, rapid, high-quality feedback about the quality of their judgments.  Unfortunately, few fields have that much quality feedback.  Kaheman et al. (2021) call practitioners with credentials but without such expert intuition &quot;respect-experts&quot;.  Kahneman (2011, p. 234) said his &quot;most satisfying and productive adversarial collaboration was with Gary Klein&quot;.&lt;/ref&gt; and 
# prefers information and sources consistent with preconceptions. ([[w:Confirmation bias|Confirmation bias]]).&lt;ref&gt;Wikipedia &quot;[[w:Confirmation bias|Confirmation bias]]&quot;, accessed 2023-04-29.&lt;/ref&gt;

To what extent do media organizations everywhere exploit the confirmation bias and overconfidence of their audience to please those who control the money for the media, and to what extent might this ''reduce'' broadly shared economic growth?  The proposed experiments should include efforts to quantify this, measuring, e.g., local incomes, inequality, political polarization and the impact of interventions attempting to improve such.  

Plous wrote, &quot;No problem in judgment and decision making is more prevalent and more potentially catastrophic than overconfidence.&quot;&lt;ref&gt;Plous (1993, p. 217).  See also Wikipedia &quot;[[w:Overconfidence effect|Overconfidence effect]]&quot;, accessed 2023-04-29.&lt;/ref&gt; It contributes to inordinate losses by all parties in negotiations of all kinds&lt;ref&gt;Thompson (2020).&lt;/ref&gt; including lawsuits,&lt;ref&gt;Loftus and Wagenaar (1988).&lt;/ref&gt; strikes,&lt;ref&gt;Babcock and Olson (1992) and Thompson and Loevenstein (1992).&lt;/ref&gt; financial market bubbles and crashes,&lt;ref&gt;Daniel et al. (1998).&lt;/ref&gt; and politics and international relations,&lt;ref&gt;&lt;!-- Dominic D.P. Johnson (2020) Strategic instincts: the adaptive advantages of cognitive biases in international politics--&gt;{{cite Q|Q120967807}}&lt;/ref&gt; including wars.&lt;ref&gt;Johnson (2004).&lt;/ref&gt;

Might the frequency and expense of lawsuits, strikes, financial market volatility, political coruption and wars be reduced by encouraging people to get more curious and search more often for information that might contradict their preconceptions?  Might such discussions be encouraged by interventions such as increasing the total funding for news through many small, independent, local news organizations?  If yes, to what extent might such experimental interventions threaten the hegemony of major media everywhere while benefitting everyone, with the possible exception of those who benefit from current systems of political corruption?  

[[File:Knowledge v. public media.png|thumb|Figure 2.  Knowledge v. public media: Percent correct answers in surveys of knowledge of domestic and international politics vs. per capita subsidies for public media in Denmark (DK), Finland (FI), the United Kingdom (UK) and the United States (US).&lt;ref&gt;&quot;politicalKnowledge&quot; dataset in Croissant and Graves (2022), originally from ch. 1, chart 8, p. 268 and ch. 4, chart 1, p. 274, McChesney and Nichols (2010).&lt;/ref&gt;]]

One attempt to quantify this appears in Figure 2, which summarizes a natural experiment on the impact of government subsidies for public media on public knowledge of domestic and international politics: Around 2008 the governments of the US, UK, Denmark and Finland provided subsidies of $1.35, $80, $101 and $101 per person per year, respectively, for public media.  A survey of public knowledge of domestic and international politics found that people with college degrees seemed to be comparably well informed in the different countries, but people with less education were better informed in the countries with higher public subsidies.  

Kaviani et al. (2022) studied the impact of &quot;the staggered expansion of [[w:Sinclair Broadcast Group|Sinclair Broadcast Group]]: the largest conservative network in the U.S.&quot; They documented a decline in Corporate Social Responsibility (CSR) ratings of firms headquartered in Sinclair expansion areas.  They also documented a &quot;right-ward ideological shift&quot; in coverage that was &quot;nearly one standard deviation of the ideology distribution&quot; as well as &quot;substantial decreases in coverage of local politics substituted by increases in national politics.&quot; Ellison (2024) said that &quot;Sinclair's recipe for TV news&quot; includes an annual survey asking viewers, &quot;What are you most afraid of?&quot; Sinclair reportedly focuses on that while implying in their coverage &quot;that America's cities, especially those run by Democratic politicians, are dangerous and dysfunctional.&quot; 

Sources in France are concerned that billionaire [[w:Vincent Bolloré|Vincent Bolloré]] has purchased a substantial portion of French media and used it effectively to promote the French far right.&lt;ref&gt;Francois (2022). Cagé (2022). Cagé and Stetler (2022).&lt;/ref&gt; Scheidler (2024a) reported that the concentration of ownership the German media &quot;has not yet reached the extreme forms observed in France, the United Kingdom or the United States, but the process of consolidation initiated several decades ago has transformed a landscape renowned for its decentralization.&quot;&lt;ref&gt;Translated from, &quot;la concentration de la propriété dans la presse suprarégionale n’a pas encore atteint les formes extrêmes observées en France, au Royaume-Uni ou aux États-Unis, mais le processus de consolidation enclenché depuis plusieurs décennies a transformé un paysage réputé pour sa décentralisation.&quot;&lt;/ref&gt;&lt;ref&gt;See also ''Die Tageszeitung'' (2023).&lt;/ref&gt; Scheidler (2024b) reported that there still exists a wide range of constructive media criticism in Germany, but it gets less coverage than before in the increasingly consolidated major media. This has driven many who are not happy with these changes to alternative media such as ''[[w:Die Tageszeitung|Die Tageszeitung]]'', founded in 1978.

Benton wrote that past research has shown that strong local newspapers &quot;increase voter turnout, reduce government corruption, make cities financially healthier, make citizens more knowledgeable about politics and more likely to engage with local government, force local TV to raise its game, encourage split-ticket (and thus less uniformly partisan) voting, make elected officials more responsive and efficient ... And ... you get to reap the benefits of all those positive outcomes ''even if you don’t read them yourself''.&quot;&lt;ref&gt;Benton (2019); italics in the original.  See also Green et al. (2023, p. 7), Schulhofer-Wohl and Garrido (2009) and Stearns and Schmidt (2022).  A not quite silly example of this is documented in the Wikipedia article on the &quot;[[w:City of Bell scandal|City of Bell scandal]]&quot; accessed 2023-05-05:  Around 1999 the local newspaper died.  In 2010 the ''[[w:Los Angeles Times|Los Angeles Times]]'' reported that the city was close to bankruptcy in spite of having atypically high property tax rates.  The compensation for the City Manager was almost four times that of the President of the US, even though Bell, California, had a population of only approximately 38,000.  The Chief of Police and most members of the City Council also had exceptionally high compensations.  It was as if the City Manager had said in 1999, &quot;Wow:  The watchdog is dead.  Let's have a party.&quot;&lt;/ref&gt;

We feel a need to repeat that last comment:  ''You and I'' benefit from others consuming news that we do not, because they become less likely to be stampeded into voting contrary to their best interests{{mdash}}and ours{{mdash}}and more likely to lobby effectively against questionable favors to major political campaign contributors or other people with power, underreported by major media that have conflicts of interest in balanced coverage of anything that might offend people with substantive control of their funding.  That suggests that everyone might benefit from subsidizing ''a broad variety of independent'' local news outlets consumed by others.&lt;ref&gt;Some of those who benefit from the current system of political corruption may lose from the increased transparency produced by increases in the quality, quantity, diversity, and broader consumption of news.  However, Bezruchka (2023) documents how even the ultra-wealthy in countries with high inequality generally have shorter life expectancies than their counterparts in more egalitarian societies:  What they might lose in social status would likely be balanced by a reduction in stress and exposure to life-threatening incidents.&lt;/ref&gt; Experiments along the lines discussed below could attempt to evaluate these claims and estimate their magnitudes.

== How fair is the US tax system? ==

How fair is the US tax system?  It depends on who is asked and how fairness is defined.  

[[File:Share of taxes vs. AGI.svg|thumb|Figure 3.  Effective tax rate vs. Adjusted Gross Income (AGI).&lt;ref&gt;York (2023) based on analyses published by the US Internal Revenue Service (IRS).&lt;/ref&gt;]]

The [[w:Tax Foundation|Tax Foundation]] computed the effective tax rate in different portions of the distribution of Adjusted Gross Income (AGI), plotted in Figure 3.  They noted that,&quot;half of taxpayers paid 99.7 percent of federal income taxes&quot;. The effective tax rate on the 1 percent highest adjusted gross incomes (AGIs) was 26 percent, almost double (1.91 times) the average, while the effective tax rate for the bottom half was 3.1 percent, only 23 percent of the average.&lt;ref&gt;York (2023).&lt;/ref&gt;

The Tax Foundation did ''not'' mention that we get a very different perspective from considering ''gross income'' rather than AGI. Leiserson and Yagan (2021)&lt;ref&gt;published by the Biden White House.&lt;/ref&gt; estimated that the average ''effective'' federal individual income tax rate paid by America’s 400 wealthiest families&lt;ref&gt;The &quot;400 wealthiest families&quot; are identified in &quot;[[w:The Forbes 400|The Forbes 400]]&quot;; see the Wikipedia article with that title, accessed 2023-05-07.&lt;/ref&gt; was between 6 and 12 percent with the most likely number being 8.2 percent. The difference comes in the ''adjustments'', while the uncertainty comes primarily from appreciation in the value of unsold stock,&lt;ref&gt;Unsold stock or other property subject to capital gains tax, which in 2022 was capped at 20 percent;  see Wikipedia, &quot;[[w:Capital gains tax in the United States|Capital gains tax in the United States]]&quot;, accessed 2023-05-08.&lt;/ref&gt; which is taxed at a maximum of 20 percent when sold and never taxed if passed as inheritance.&lt;ref&gt;The Wikipedia article on &quot;[[w:Estate tax in the United States|Estate tax in the United States]]&quot; describes an &quot;Exclusion amount&quot;, which is not taxed in inheritance.  That exclusion amount was $675,000 in 2001 and has generally trended upwards since except for 2010, and was $12.06 million in 2022. (accessed 2023-05-08.)&lt;/ref&gt;

Divergent claims about ''business'' taxes can similarly be found.  Watson (2022) claimed  that, &quot;Corporate taxes are one of the most economically damaging ways to raise revenue and are a promising area of reform for states to increase competitiveness and promote economic growth, benefiting both companies and workers.&quot;  His &quot;economically damaging&quot; claim seems contradicted by the claim of Piketty (2021, p. 139), cited with Figure 1 above, that when the top income tax rate was cut in half, the rate of economic growth in the US fell by half, instead of increasing as Watson (2022) suggested. By contrast, Fuhrmann and Uradu (2023) describe, &quot;How large corporations avoid paying taxes&quot;.   

[[File:UStaxWords.svg|thumb|Figure 4.  Millions of words in the US federal tax code and regulations, 1955-2015, according to the [[w:Tax Foundation|Tax Foundation]]. [1=income tax code; 2=other tax code; 3=income tax regulations; 4=other tax regulations; solid line= total]&lt;ref&gt;&quot;UStaxWords&quot; dataset in Croissant and Graves (2022) from the Tax Foundation.&lt;/ref&gt;]]

One reference on the difference between &quot;adjusted&quot; and &quot;gross&quot; income is US federal tax code and regulations, which grew from 1.4 million words in 1955 to over 10 million in 2015, averaging 145,000 additional words each year;  see Figure 4.  

How does this relate to media?

== How do media organizations make money? ==

Media organizations everywhere sell changes in audience behaviors to the people who give them money.  If they do not have an audience, they have nothing to sell.  If they sufficiently offend their funders, they will not get the revenue needed to produce content.&lt;ref&gt;A famous illustration of this conflict between content and funding was when CBS Chairman [[w:William S. Paley|William Paley]] reportedly told [[w:Edward R. Murrow|Edward R. Murrow]] in 1958 that he was discontinuing Murrow's award-winning show ''[[w:See It Now|See It Now]]'', because &quot;I don't want this constant stomach ache every time you do a controversial subject&quot;, documented in Friendly (1967, p. 92).&lt;/ref&gt;  

The major media in the US have conflicts of interest in honestly reporting on discussions in congress on copyright law or on anything that might impact a major advertiser or might make it easier for politicians to get elected by spending less money on advertising.  McChesney (2015) insisted that the major media are not interested in providing information that people want:  They are interested in making money and protecting the interests of the ultra-wealthy, who control the largest advertising budgets.  For example, media coverage of the roughly 40,000 people who came to [[w:1999 Seattle WTO protests|Seattle in 1999 to protest the WTO]] Ministerial Conference there&lt;ref&gt;Wikipedia &quot;[[w:1999 Seattle WTO protests|1999 Seattle WTO protests]]&quot;, accessed 2023-05-08.&lt;/ref&gt; and the 10,000 - 15,000 who came to [[w:Washington A16, 2000|Washington, DC, the following year]] to protest the International Monetary Fund and the World Bank,&lt;ref&gt;Wikipedia &quot;[[w:Washington A16, 2000|Washington A16, 2000]]&quot;, accessed 2023-05-08.&lt;/ref&gt; included &quot;some outstanding pieces produced by the corporate media, but those were exceptions to the rule. ... [T]he closer a story gets to corporate power and corporate domination of our society, the less reliable the corporate news media is.&quot;&lt;ref&gt;McChesney (2015, p. xx).&lt;/ref&gt; 

Aaron (2021) said, &quot;Bob McChesney ... taught me [to] look at ... the stories that are cheap to cover.&quot; Between around 1975 and 2000, the major commercial broadcasters in the US fired nearly all their investigative journalists&lt;ref&gt;McChesney (2004, p. 81):  &quot;A five-year study of investigative journalism on TV news completed in 2002 determined that investigative journalism has all but disappeared from the nation's commercial airwaves.&quot;&lt;/ref&gt; and replaced them with the police blotter.  It's easy and cheap to repeat what the police say.&lt;ref&gt;Holmes (2022) quoted Ryan Sorrell, Founder and Publisher of the Kansas City Defender, as saying, &quot;the media often parrots or repeats what police and news releases say.&quot;&lt;/ref&gt; A news outlet can do that without seriously risking loss of revenue.  In addition, poor defendants who may not have money for legal defense will rarely have money to sue a media outlet for defamation.  By contrast, a news report on questionable activities by a major funder risks both direct loss of advertising revenue and being sued.&lt;ref&gt;The risks of being sued include the risks of [[w:Strategic lawsuit against public participation|strategic lawsuits against public participation]] (SLAPPs) by major organizations, which can intimidate journalists and publishers as well as potential whistleblowers, who might inform journalists of violations of law by their employers.  Some of these are documented in the &quot;[[w:Freedom of the press in the United States#U.S. Press Freedom Tracker|U.S. Press Freedom Tracker]]&quot;, maintained by the [[w:Freedom of the Press Foundation|Freedom of the Press Foundation]] and the [[w:Committee to Protect Journalists|Committee to Protect Journalists]].  These include arrests, assaults, threats, denial of access, equipment damage, prior restraint, and subpoenas which could intimidate journalists, publishers, and employees feeling a need to expose violations of law and threats to public safety.  See Wikipedia &quot;[[w:Freedom of the Press Foundation|Freedom of the Press Foundation]]&quot;, &quot;[[w:Committee to Protect Journalists|Committee to Protect Journalists]]&quot;, and &quot;[[w:Strategic lawsuit against public participation|Strategic lawsuit against public participation]]&quot;, accessed 2023-07-11.&lt;/ref&gt;  These risks impose a higher standard of journalism (and additional costs) when reporting on questionable activities by people with power than when reporting on poor people.  This is a much bigger problem in countries where libel is a criminal rather than a civil offense or where truth is not a defense for libel.&lt;ref&gt;Islam et al. (2002), esp. pp. 12-13 (24-25/336 in pdf), p. 50 (62/336 in pdf), and ch. 11, pp. 207-224 (219-236/336 in pdf). [[w:United States defamation law|Truth was not a defense against libel in the US]] in 1804 when Harry Croswell lost in ''[[w:United States defamation law#People v. Croswell|People v. Croswell]]''.  That began to change the next year when the [[w:United States defamation law#People v. Croswell|New York State Legislature]] changed the law to allow truth as a defense against a libel charge. Seventy years earlier in 1735 [[w:John Peter Zenger#Libel case|John Peter Zenger]] was acquitted of a libel charge, but only by [[w:Jury nullification in the United States|jury nullification]].&lt;/ref&gt;

[[File:U.S. incarceration rate since 1925.svg|thumb|Figure 5.  Percent of the US population in state and federal prisons  [male (dashed red), combined (solid black), female (dotted green)]&lt;ref&gt;&quot;USincarcerations&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

After about 1975 the public noticed the increased coverage of crime in the broadcast news and concluded that crime was out of control, when there had been no substantive change in crime.  They voted in a generation of politicians, who promised to get tough on crime.  The incarceration rate in the US went from 0.1 percent to 0.5 percent in the span of roughly 25 years, after having been fairly stable for the previous 50 years;  see Figure 5.&lt;ref&gt;Potter and Kapeller (1998). Sacco (1998, 2005).&lt;/ref&gt;  

[[File:IncomeInequality9b.svg|thumb|Figure 6.  Average and quantiles of family income (Gross Domestic Product per family) in constant 2010 dollars.&lt;ref&gt;&quot;incomeInequality&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

Around that same time, income inequality in the US began to rise;  see Figure 6.&lt;ref&gt;Bezruchka (2023) summarizes research documenting how &quot;inequality kills us all&quot;. He noted that the US was among the leaders in infant mortality and life expectancy in the 1950s.  Now the US is trailing most of the advanced industrial democracies per United Nations (2022).  He attributes the slow rate of improvement in public health in the US to increases in inequality.  That argument is less than perfect, because Figure 5 suggests that inequality in the US did not begin to increase until around 1975, but the divergence in public health between the US and other advanced industrial democracies seems more continuous between the 1950s and the present, 2023.  Beyond this, Graves and Samuelson (2022) noted that it is in everyone's best interest to help others with conditions that might be infectious to get competent medical assistance, because that reduces our risk of contracting their disease and possibly dying from it. Bezruchka (2023) cites documentation claiming that even the wealthy in the US have lower life expectancy than their counterparts in other advanced industrial democracies, because the high level of inequality in the US means that the ultra-wealthy in the US get exposed to more pathogens than their counterparts elsewhere.  See also Wilkinson and PIckett (2017).&lt;/ref&gt; To what extent might that increase in inequality be due to the structure of the major media?  

To what extent might you and I benefit from making it easier for millions of others to research different aspects of government policies including the &quot;adjustments&quot; in the US tax system embedded in the over 10 million words of US federal tax code and regulations documented with Figure 4 above, encouraging them to lobby the US Congress against the special favors granted to major political campaign contributors against the general welfare of everyone else? Everyone except the beneficiaries of such political corruption would likely benefit from the news that helps concerned citizens lobby effectively against such corruption, even if we did not participate in such citizen lobbying efforts and were completely ignorant of them.

== Media and war == 

:[[w:You've Got to Be Carefully Taught|''&quot;You've got to be taught to hate and fear. ... It's got to be drummed in your dear little ear.&quot;'']] 
:Lt. Cable in the [[w:South Pacific (musical)|1949 Rodgers and Hammerstein musical ''South Pacific'']].  

To what extent is it accurate to say that before anyone is killed in armed hostilities, the different parties to the conflict are polarized by the different media the different parties find credible?&lt;ref&gt;The role of the media in war has long been recognized.  It is commonly said that the first casualty of war is truth. Knightley (2004, p. vii) credits Senator Hiram Johnson as saying in 1917, &quot;The first casualty when war comes is truth.&quot; However, the Wikiquote article on &quot;[[Wikiquote:Hiram Johnson|Hiram Johnson]]&quot; says this quote has been, &quot;Widely attributed to Johnson, but without any confirmed citations of original source. ... [T]he first recorded use seems to be by Philip Snowden.&quot; (accessed 2023-07-22.)&lt;/ref&gt; 

This might seem obvious, but how can we quantify political polarization in a way (a) that correlates with the severity of the conflict and (b) can be used to evaluate the effectiveness of efforts to reduce the polarization? 

The [[w:International Panel on the Information Environment|International Panel on the Information Environment]] (IPIE) is a consortium of over 250 global experts developing tools to combat political polarization driven by the structure of the Internet.&lt;ref&gt;e.g., National Acadamies (2023).&lt;/ref&gt; The US Institute of Peace (2016) discusses &quot;Tools for Improving Media Interventions in Conflict Zones&quot;. Previous research in this area was summarized by Arsenault et al. (2011). One such tool might be video games.&lt;ref&gt;Caelin (2016).&lt;/ref&gt;

We suggest experimenting with interventions designed to reduce political polarization with some of the smallest but most intense conflicts: Interventions that require money could be more effectively tested with smaller, high intensity conflicts. With randomized controlled trials, it would be easier to measure a reduction from a higher-intensity conflict, and a smaller population could commit more money per capita with a relatively modest budget.

The [[w:Armed Conflict Location and Event Data Project|Armed Conflict Location and Event Data Project (ACLED)]] tracks politically relevant violent and nonviolent events by a range of state and non-state actors. Their data can help identify countries or geographic regions in conflict as candidates to be [[w:Randomized controlled trial|assigned randomly to experimental and control groups]], whose comparison can provide high quality data to help evaluate the impact of any intervention. Initial experiments of this nature might be done with a modest budget by working with organizations advocating nonviolence and with religious groups to recruit diaspora communities to do things recommended by experts in IPIE while also lobbying governments for funding. Any success can be leveraged into changes in foreign and military policies to make the world safer for all. 

Before discussing such experiments further, we consider a few case studies. 

=== Russo-Ukrainian War and the US Civil War === 

In the [[w:Russo-Ukrainian War|Russo-Ukrainian War]], Halimi and Rimbert (2023) describe &quot;Western media as cheerleaders for war&quot;. [[w:Joseph Stiglitz|Stiglitz]] (2002) noted this was a general phenomenon:  &quot;In periods of perceived conflict ... a combination of self-censorship and reader censorship may also undermine the ability of a supposedly free press to ensure democratic transparency and openness.&quot; Media organizations do not always do this solely to please their funders.  Reporters are killed&lt;ref&gt;Different lists of journalists killed for their work are maintained by the [[w:Committee to Protect Journalists|Committee to Protect Journalists]], (CPJ), [[w:Reporters Without Borders|Reporters without Borders]], and the [[w:International Federation of Journalists|International Federation of Journalists]].  CPJ has claimed that their numbers are typically lower, because their confirmation process may be more rigorous.  See Committee to Protect Journalists (undated) and the Wikipedia articles on &quot;[[w:Committee to Protect Journalists|Committee to Protect Journalists]]&quot;, &quot;[[w:Reporters Without Borders|Reporters without Borders]]&quot;, and the &quot;[[w:International Federation of Journalists|International Federation of Journalists]]&quot;, accessed 2023-07-11.&lt;/ref&gt; or jailed and news outlets closed to prevent them from disseminating information that people with power do not want distributed.  Early in the Civil War in the US (1861-1865), some newspapers in the North said the US should let the South secede, because that would be preferable to war.  Angry mobs destroyed some offices and printing presses.  One editor &quot;was forcibly taken from his house by an excited mob, ... covered with a coat of tar and feathers, and ridden on a rail through the town.&quot; Others changed their policies &quot;voluntarily&quot;, recognizing threats to their lives or property or to a loss of audience.&lt;ref&gt;Harris (1999, esp. p. 100).&lt;/ref&gt;

=== Hitler ===

Fulda (2009) studied the co-evolution of newspapers and party politics in Germany, focusing primarily on Berlin, 1924-1933. During that period, the [[w:Nazi Party|Nazis (NSDAP, National Socialist German Workers' Party, Nationalsozialistische Deutsche Arbeiterpartei)]] grew from 2.6 percent of the votes for the [[w:Reichstag (Nazi Germany)|Reichstag (German parliament)]] in 1928 to 44 percent in 1933. Fulda described exaggerations in the largely tabloid press of an indecisive government incapable of managing either the economy or the increasing political violence, blamed excessively on Communists, and the potential for civil war. This turned the Nazis into an attractive choice for voters desperate for decisive action.&lt;ref&gt;Fulda (2009, Abstract, ch. 6, &quot;War of Words: The Spectre of Civil War, 1931–2&quot;.&lt;/ref&gt; 

After the 1933 elections, the Reichstag passed the &quot;[[w:Enabling Act of 1933|Enabling Act of 1933]], which gave Hitler's cabinet the right to enact laws without the consent of parliament. The Nazis then began full censorship of the newspapers, physically beating, imprisoning and in some cases killing journalists, as the leading publishers acquiesced. The primary sources of news during that period was newspapers; German radio was relatively new during that period and carried very little news. Most newspapers were [[w:Tabloid journalism|tabloids]], interested in either making money or promoting a party line with minimal regard for fact checking. A big loser in this was the right‐wing press magnate [[w:Alfred Hugenberg|Alfred Hugenberg]], whose political mismanagement led to the substantial demise of his [[w:German National People's Party|German National People's Party (Deutschnationale Volkspartei, DNVP)]], mostly benefitting Hitler.&lt;ref&gt;Fulda (2009).&lt;/ref&gt; 

This suggests the need for a [[w:Counterfactual history|counterfactual analysis of this period]], asking what kinds of changes in the structure of the media ecology might have prevented the rise of the Nazis? In particular, to what extent might a more diverse local news environment supported by citizen-directed subsidies as suggested herein have reduced the risk of a demise of democracy? And might some sort of [[w:Fairness Doctrine|fairness doctrine]] have helped?&lt;ref name=fairness&gt;Wikipedia &quot;[[w:FCC fairness doctrine|FCC fairness doctrine]]&quot;, accessed 2023-07-21.&lt;/ref&gt; And how might different rules for distributing different levels of funding to local news outlets impact the level of democratization? (Threats to democracy include legislation like the German Enabling Act of 1933 and other situations that allow an executive to successfully ignore the will of an otherwise democratic legislature, a [[w:self-coup|self-coup]], as well as a military coup.)

=== Stalin and Putin === 

[[File:Russian economic history 1885-2018.svg|thumb|Figure 7. Gross Domestic Product per person in Russian 1885-2018 in thousands of 2011 dollars]]

A 2017 survey asking Russians to name 10 of the world’s most prominent personalities listed Joe Stalin and Vladimir Putin as the top two with 38 and 34 percent, respectively. When the study was redone in 2021, Putin had slipped from number 2 to number 5. Stalin still led with 39 percent followed by Vladimir Lenin with 30 percent, Poet Alexander Pushkin and tsar Peter the Great with 23 and 19 percent each, then Putin with 15 percent.&lt;ref&gt;&lt;!-- Putin Plummets, Stalin Stays on Top in Russians’ Ranking of ‘Notable’ Historical Figures – Poll--&gt;{{cite Q|Q123197680}} &lt;!-- The most outstanding personalities in history (in Russian) --&gt;{{cite Q|Q123197317}}&lt;/ref&gt; 

It may be difficult for some people in the West to understanding how Stalin and Putin could be so popular, given the way they have been typically described in the mainstream Western media.&lt;ref&gt;[[w:Joseph Stalin|Joseph Stalin]] got much better press in the US during the Great Depression and World War II than he has gotten since 1945.&lt;/ref&gt; However, this is relatively easy to understand just by looking at the accompanying plot (Figure 7) of average annual income in that part of the world between 1885 and 2018: Both Stalin and Putin inherited economies that had fallen dramatically in the previous years and supervised dramatic improvements. Putin's decline between 2017 and 2021 may also be understood from this plot, because it shows how the dramatic growth that began around the time that Putin became acting President of Russia has slowed substantially since 2012. 

Similar comments could be made about the Vietnam war and the &quot;War on Terror&quot;.&lt;ref&gt;The Wikiversity article on &quot;[[Winning the War on Terror]]&quot; discusses the role of the media in the &quot;War on Terror&quot; and other conflicts including Vietnam.&lt;/ref&gt; To what extent can the experiments described in this article contribute to understanding the role of the media in stoking hate and fear, and how that might be impacted by citizen-direct subsidies for more and more diverse local media?

=== Iraq and the Islamic State ===

[[w:Fall of Mosul|In 2014 in Mosul, two Iraqi army divisions totaling 30,000 and another 30,000 federal police]] were overwhelmed in six days by roughly 1,500 committed Jihadists. Four months later, ''Reuters'' reported that, &quot;there were supposed to be close to 25,000 soldiers and police in the city; the reality ... was at best 10,000.&quot; Many of the missing 15,000 were &quot;ghost soldiers&quot; kicking back half their salaries to their officers. Also, &quot;[i]nfantry, armor and tanks had been shifted to Anbar, where more than 6,000 soldiers had been killed and another 12,000 had deserted.&quot;&lt;ref&gt;Parker et al. (2014).&lt;/ref&gt; 

To what extent might the political corruption and low moral documented in that ''Reuters'' report have been allowed to grow to that magnitude if Iraq had had a vigorous adversarial press, as discussed in this article? Instead, Paul Bremer, who was appointed as the [[w:Paul Bremer#Provisional coalition administrator of Iraq|Provisional coalition administrator of Iraq]] just over a week after President George W. Bush's [[w:Mission Accomplished speech|Mission Accomplished speech]] of 2003-05-01, imposed strict press censorship.&lt;ref&gt;McChesney and Nichols (2010, p. 242).&lt;/ref&gt; McChesney and Nichols contrasted this with General Eisenhower, who &quot;called in German reporters [after the official surrender of Nazi Germany in WW II] and told them he wanted a free press. If he made decisions that they disagreed with, he wanted them to say so in print.&quot;&lt;ref&gt;McChesney and Nichols (2010, Appendix II. Ike, MacArthur and the Forging of Free and Independent Press, pp. 241-254).&lt;/ref&gt;

=== Israel-Palestine ===

::''Those who make peaceful revolution impossible will make violent revolution inevitable.''
:::-- John F. Kennedy (1962)

To what extent is the [[w:Israeli–Palestinian conflict|Israeli–Palestinian conflict]] driven by differences in the media consumed by the different parties to that conflict? 

* To what extent are the supporters of Israel aware of violent acts committed by Palestinians but are ''unaware'' of the actions by Israelis that have motivated those violent acts? 
* Similarly, to what extent are the supporters of Palestinians unaware of or downplay the extent to which violence by Palestinians motivate the actions of Israel against them? 

To what extent are these differences in perceptions between supporters of Israel and supporters of Palestinians driven by differences in the media each find credible? 

What can be done to bridge this gap?  

[[w:Gene Sharp|Gene Sharp]], [[w:Mubarak Awad|Mubarak Awad]], and other supporters of [[w:nonviolence|nonviolence]] have suggested that when nonviolent direct action works, it does so by exposing a gap between the rhetoric [supported by the major media] and the reality of their opposition. Over time, this gap erodes pillars of support of the opposition. One example was the nonviolence of the [[w:First Intifada|First Intifada]] (1987-1993), which were protests against &quot;beatings, shootings, killings, house demolitions, uprooting of trees, deportations, extended imprisonments, and detentions without trial.&quot;&lt;ref&gt;Ackerman and DuVall (2000, p. 407).&lt;/ref&gt; As that campaign began, Israel got so much negative press for killing nonviolent protestors, that Israeli Defense Minister [[w:Yitzhak Rabin|Yitzhak Rabin]] ordered his soldiers NOT to kill but instead to shoot to wound. As the negative press continued, he issued wooden and metal clubs with orders to break bones.&lt;ref&gt;Shlaim (2014).&lt;/ref&gt; As the negative press still continued, Rabin ran for Prime Minister on a platform of negotiating with Palestinians. His victory and subsequent negotiations led to the [[w:Oslo Accords|Oslo Accords]] and the joint recognition of each other by the states of Israel and [[w:State of Palestine|Palestine]]. The West Bank and Gaza have continued under Israeli occupation since with some services provided by the official government of Palestine. 

During the Intifada, Israel tried to infiltrate the protestors with [[w:agent provocateur|agents provocateurs]] in Palestinian garb. They were exposed and neutralized until Israel deported 481 people they thought were leading the nonviolence who were accepted in other countries and imprisoned tens of thousands of others suspected of organizing the nonviolence. Finally, they got the violence needed to justify a massively violent repression of the Intifada.&lt;ref&gt;King (2007).&lt;/ref&gt;

The general thrust of this current analysis suggests a two pronged intervention to reduce the risk of a continuation of the violence that has marked Israel-Palestine since at least 1948: 
# Offer nonviolence training to all Palestininans, Israelis and supporters of either interested in the topic. This is the opposite of the policies Israel pursued during the First Intifada, at least according to the references cited in this discussion of that campaign.&lt;ref&gt;It also is the opposite of the decision of the US Supreme Court in ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', which ruled that teaching nonviolence to someone designated as a terrorist was a crime under the [[w:Patriot Act|Patriot Act]], as it provided &quot;material support to&quot; a foreign terrorist organization.&lt;/ref&gt;
# Provide citizen-direct subsidies to local news nonprofits in the West Bank and Gaza at, e.g., 0.15 percent of GDP, as recommended by McChesney and Nichols, cited above. 

How can we evaluate the budget required for such an experiment? The nominal GDP of the [[w:State of Palestine|State of Palestine]] in 2021 was estimated at $18 billion; 0.15% of that is $27 million. Add 10% for research to get $30 million per year. That ''annual'' cost for the media component of this proposed intervention is 12% of the billion Israeli sheckels ($246 million) that the Gaza war was costing Israel ''each day'' in the early days of the [[w:Israel-Hamas war|Israel-Hamas war]], according to the Israeli Finance Minister on 2023-10-25.&lt;ref&gt;Reuters (2023-10-25).&lt;/ref&gt; As this is being written, that war has continued for over 100 days. If the average daily cost of that war to Israel during that period has been $246 million, then that war will have already cost Israel over $24.6 billion. And that does not count the loss of lives and the destruction of property in Gaza and the West Bank. 

How much would training in nonviolence cost? That question would require more research, but if it were effective, the budget would seem to be quite modest compared to the cost of war, even if it were several times the budget for citizen-directed subsidies for local news in Palestine as just suggested.

== The decline of newspapers == 

[[File:Newspapers as a percent of US GDP.svg|thumb|Figure 8.  US newspaper revenue 1955-2020 as a percent of GDP.&lt;ref&gt;&quot;USnewspapers&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

McChesney and Nichols (2022) noted that US newspaper revenue as a percent of GDP fell from over 1 percent in 1956 to less than 0.1 percent in 2020;  see Figure 8.  Abernathy (2020) noted that the US lost more than half of all newspaper journalists between 2008 and 2018.&lt;ref&gt;Abernathy (2020, p. 22).&lt;/ref&gt; A quarter of US newspapers closed between 2004 and 2020,&lt;ref&gt;Abernathy (2020, p. 21).&lt;/ref&gt; and many that still survive are publishing less, creating &quot;news deserts&quot; and &quot;ghost newspapers&quot;, some with no local journalists on staff.&lt;ref&gt;Abernathy (2020) documented the problem of increasing &quot;news deserts and ghost newspapers&quot; in the US. A local jurisdiction without a local news outlet has been called a &quot;news desert&quot;.  She uses the term &quot;ghost newspapers&quot; to describe outlets &quot;with depleted newsrooms that are only a shadow of their former selves.&quot;  Some “ghost newspapers” continue to publish with zero local journalists, produced by reporters and editors that don't live there. One example is the ''Salinas Californian'', a 125-year-old newspaper in Salinas, California, which lost its last paid journalist 2022-12, according to the Los Angeles Times (2023-03-27).  They continue to publish, though &quot;The only original content from Salinas comes in the form of paid obituaries, making death virtually the only sign of life at an institution once considered a must-read by many Salinans.&quot; A leading profiteer in this downward spiral is reportedly [[w:hedge fund|hedge fund]] [[w:Alden Global Capital|Alden Global Capital]].  Threisman (2021) reported that, &quot;When this hedge fund buys local newspapers, democracy suffers&quot;.  And Benton (2021) said, &quot;The vulture is hungry again: Alden Global Capital wants to buy a few hundred more newspapers&quot;. Hightower (2023) describes two organizations fighting this trend.  One is National Trust for Local News, a nonprofit that recently bought several local papers and &quot;is turning each publication over to local non-profit owners and helping them find ways to become sustainable.&quot; The other is CherryRoad Media, which &quot;bought 77 rural papers in 17 states, most from the predatory Gannett conglomerate that wanted to dump them&quot;, and is working to &quot;return editorial decision-making to local people and journalists ... and ... reinvest profits in real local journalism that advances democracy.&quot; News outlets acquired by something like the National Trust for Local News should be eligible for citizen-directed subsidies for local news, as discussed below, after their ownership was officially transferred to local humans.  Outlets acquired by organizations like CheeryRoad Media would not be eligible as long as they remained subsidiaries.&lt;/ref&gt; More recent news continues to be dire. The Fall 2023 issue of ''Columbia Journalism Review'' reported that 2023 &quot;has become media’s worst year on record for job losses&quot;.&lt;ref&gt;Columbia Journalism Review (2023).&lt;/ref&gt; Substantial advertising revenue has shifted to the &quot;click economy&quot;, where advertisers pay for clicks, especially on social media.&lt;ref&gt;Carter (2021).&lt;/ref&gt;

Newpapers in other parts of the world have also experienced substantial declines in revenue. In 2013 German law was changed to inclued &quot;[[w:Ancillary copyright for press publishers|Ancillary copyright for press publishers]]&quot;, also called a &quot;link tax&quot;. However, this law was declared invalid in 2019 the European Court of Justice (ECJ), because it had been submitted in advance to the [[w:European Commission|EU Commission]], as required.&lt;ref&gt;&lt;!--Axel Kannenberg (2019) ECJ: German ancillary copyright law invalid for publishers, heise online--&gt;{{cite Q|Q124051681|title= ECJ: German ancillary copyright law invalid for publishers}}&lt;/ref&gt; Before that ECJ decision, Google had removed newspapers from Google News in Germany. German publishers then reached an agreement with Google after traffic to their websites plummeted.&lt;ref&gt;&lt;!--Dominic Rushe (2014) Google News Spain to close in response to story links 'tax', Guardian--&gt;{{cite Q|Q124051847}}&lt;/ref&gt; Building on that and similar experience in Spain, the European Union adopted a [[w:Directive on Copyright in the Digital Single Market|Directive on Copyright in the Digital Single Market]] in 2019. A similar link tax proposal in Canada led [[w:Meta|Meta]], the parent company of Facebook, to withdraw news from Canada, and Google agreed to 'pay about $100 million a year into a new fund to support “news&quot;' in Canada. As of 2023-11-30, California was still considering a link tax.&lt;ref&gt;&lt;!--Ken Doctor (2023) Forget the link tax. Focus on one key metric to “save local news, NiemanLab--&gt;{{cite Q|Q124051930}}&lt;/ref&gt;

== Threats from social media == 

The growth of social media has been wonderful and terrible.  It has been wonderful in making it easier for people to maintain friendships and family ties across distances.&lt;ref&gt;Friedland (2017) noted that the Internet works well at the global level, helping people get information from any place in the world, and at the micro level, e.g., with Facebook helping people with similar diseases find one another.  It does not work well at the '“meso level arenas of communication” in the middle.  They're not big enough to aggregate all the scale that goes into creating a worldwide web or even a Wikipedia. See also Lloyd and Friedland (2016).&lt;/ref&gt; 

But it has also been terrible as  &quot;antisocial media&quot;&lt;ref&gt;Vaidhyanathan (2018).&lt;/ref&gt; have been implicated in the relatively recent rise in dysfunctional and counterproductive political polarization and violence. Ding et al. (2023) document, &quot;Same words, different meanings&quot; in their use by [[w:CNN|CNN]] and [[w:Fox News|Fox News]] and how that has interacted with word usage on Twitter between 2010 and 2020 to increase political polarization, &quot;impeding rather than supporting online democratic discourse.&quot;&lt;ref&gt;See also Ashburn.&lt;/ref&gt; 

Extreme examples of this increase have included violent efforts to prevent peaceful transitions of power in the US&lt;ref&gt;Wikipedia &quot;[[w:January 6 United States Capitol attack|January 6 United States Capitol attack]]&quot;, accessed 2023-05-09.&lt;/ref&gt; and Brazil.&lt;ref&gt;Wikipedia &quot;[[w:2023 Brazilian Congress attack|2023 Brazilian Congress attack]]&quot;, accessed 2023-05-09.&lt;/ref&gt; These changes even threaten the national security of the US and its allies,&lt;ref&gt;McMaster (2020). Zuboff (2019) noted that data on many aspects of ordinary daily life are captured and used by people with power for various purposes.  For example, data on people's locations captured from their mobile phones are used to try to sell them goods and services.  Data on a child playing with a smart Barbie doll are used to inculcate shopping habits in child and caregiver.  If you are late on a car payment, your keys can be deactivated until a tow truck can arrive to haul it away.  To what extent do the major media today have conflicts of interest in honestly reporting on this?  How might the experiments proposed herein impact the commercial calculus of major media and the political economy more generally?&lt;/ref&gt; according to [[w:H. R. McMaster|H. R. McMaster]],&lt;ref&gt;Wikipedia &quot;[[w:H. R. McMaster|H. R. McMaster]]&quot;, accessed 2023-05-09.&lt;/ref&gt; President Trump's second national security advisor. 

Various responses to these concerns have been suggested, beyond the recommendations of McChensey and Nichols.  These include the following:&lt;ref&gt;See also the section on &quot;&quot;[[International Conflict Observatory#Suggested responses to these concerns|Suggested reponses to these concerns]]&quot; in the Wikiversity article on &quot;[[International Conflict Observatory]]&quot;.&lt;/ref&gt;:

* Make internet companies liable for defamation in advertisements, similar to print media and broadcasting.&lt;ref&gt;See Baker (2020) and the Wikiversity article on &quot;[[Dean Baker on unrigging the media and the economy]], accessed 2023-07-26.&lt;/ref&gt;  
* Tax advertising revenue received by large internet companies and use that to fund more local media.&lt;ref&gt;Karr and Aaron (2019).&lt;/ref&gt; 
* Replace advertising as the source of funding for social media with subscriptions.&lt;ref&gt;Frank (2021) wrote, &quot;[D]igital aggregators like Facebook ... make money not by charging for access to content but by displaying it with finely targeted ads based on the specific types of things people have already chosen to view. If the conscious intent were to undermine social and political stability, this business model could hardly be a more effective weapon. ... [P]olicymakers’ traditional hands-off posture is no longer defensible.&quot;&lt;/ref&gt; 

To these suggestions, we add the following:  

* Allow some of but not all citizen-directed subsidies for news to go to social media outlets, as suggested below.  
* Require that all organizations whose income depends on promoting or &quot;boosting&quot; content, whether in advertisements or &quot;underwriting spots&quot; or [[w:clickbait|clickbait]], to provide copies of the ads, underwriting spots and clickbait to a central repository like the [[w:Internet Archive|Internet Archive]].  
* Use advertising to discuss overconfidence and encourage people to talk politics with humility and respect, recognizing that the primary differences they have with others are the media they consume.&lt;ref&gt;For studies of ad campaigns in other contexts, see Piwowarski et al. (2019) and Tom-Yov (2018), cited above in discussing &quot;Reducing political polarization&quot;.&lt;/ref&gt;

== How to counter political polarization ==

More research seems to be needed on how to counter the relatively recent increases in political polarization. For example, might some form of [[w:Fairness doctrine|fairness doctrine]]&lt;ref name=fairness/&gt; help reduce political polarization? [[w:Fairness doctrine#Opposition|Conservative leaders are vehemently opposed]], insisting it would be an attack on First Amendment rights. However, as noted above, the tabloid media of Germany seems to have contributed to Hitler's rise to power between 1924 and 1933. How is the increase in political polarization since 1987 and 2004 different from the disregard for fairness of the news media that helped bring Hitler to power? One example: The lawsuit ''[[w:Dominion Voting Systems v. Fox News Network|Dominion Voting Systems v. Fox News Network]]'' was settled with Fox agreeing to to pay Dominion $787.5 million while acknowledging that Fox had knowingly and intentionally made false and defamatory statements about Dominion to avoid losing audience to media outlets that continued to claim fraudulently that Donald Trump not Joe Biden had won the 2020 US presidential election. The settlement permitted Fox to avoid apologizing publicly, which could have threatened their audience share. That settlment was less than 6 percent of Fox's 2022 revenue of $14 billion.&lt;ref&gt;&lt;!--Fox earnings release for the quarter and fiscal year ended June 30, 2022--&gt;{{cite Q|Q124003735}}&lt;/ref&gt; Evidently, ''if that decision made a difference of 6 percent in their audience ratings, Fox made money from defaming Dominion even after paying them $787.5 million.'' If so, it was a good business decision, especially since they did not have to publicly apologize.  

To what extent did Fox's lies about Dominion contribute to the [[w:January 6 United States Capitol attack|mob attacks on the US Capitol on January 6, 2021]], trying to prevent the US Congress from officially declaring that Joe Biden had won the 2020 elections? And what are elected officials prepared to do to improve understanding of what contributes to increases in political polarization and how political differences can be made less lethal and more productive?

== McChesney and Nichols' Local Journalism Initiative ==

As noted above, McChesney and Nichols (2021, 2022) propose a &quot;Local Journalism Initiative&quot;, distributing 0.15 percent of GDP to local news nonprofits via local elections.  They based this partly on their earlier work suggesting that subsidies for newspapers in the US in 1840 was around 0.2 percent of GDP.&lt;ref&gt;McChesney and Nichols (2010, 2016).&lt;/ref&gt;

=== McChesney and Nichols' eligibility criteria === 

To be eligible, McChesney and Nichols say the recipient of such funds should satisfy the following:&lt;ref&gt;McChesney and Nichols (2021, 2022).  They also suggest having the US Postal Service administer this with elections every three years.&lt;/ref&gt; 
* Be a local nonprofit with at least six months of history, so voters could know their work.  
* Be locally based with at most 75 percent of salaries going to local residents.  
* Be completely independent, not a subsidiary of a larger organization.
* Produce and publish original material at least five days per week on their website for free, explicitly in the public domain. 
* Each voter is asked to vote for at least three different local news outlets to support diversity. 
* No single news outlet should get more than 25 percent of that jurisdiction's annual budget for local news subsidies.  
* Each recipient of these subsidies should get at least 1 percent of the vote to qualify, or 0.5 percent of the vote in political jurisdictions with over 1 million people. Diversity and competition are crucial.  
* There will be no content monitoring:  Government bureaucrats will not be allowed to decide what is &quot;good journalism&quot;.  That's up to the voters.  
* Voting would be limited to those 18 years and older.  

=== Alternatives === 

Some aspects of this might be relaxed for at least some political jurisdictions included in an experiment.  For example, might it be appropriate to allow for-profit news outlets to compete for these subsidies as long as they meet the other criteria?&lt;ref&gt;Kaiser (2021) noted that nonprofits in the US cannot endorse political candidates and are limited in how they can get involved in debates on political issues.  Do restrictions like these contribute to the general welfare?  Or might the public interest be better served with citizen-directed subsidies for media that might be more partisan?  This is one more question that might be answered by appropriate experimentation.&lt;/ref&gt; However, we prefer to retain the rules requiring recipients to be local and completely independent, at least for many experimental jurisdictions.&lt;ref&gt;Various contributors to Islam et al., eds. (2002) raised questions about concentrations of power in large media organizations, especially Herman, ch. 4, pp. 61-81 (73-97/336 in pdf).  Djankov et al. (2002) found that &quot;Government ownership of the media is detrimental to economic, political, and-most strikingly-social outcomes&quot;, including education and health.&lt;/ref&gt; If citizen-directed subsidies for local news go to for-profit organizations, to what extent should their finances be transparent, e.g., otherwise complying with the rules for 501(c)(3)s? Might it also be appropriate to allow some portion of these funds to be distributed to noncommercial ''social media'' outlets that submitted all their content to a public, searchable database like the [[w:Internet Archive|Internet Archive]]?   

News written by people paid with these subsidies should be available under a free license like Creative Commons Attribution-ShareAlike (CC BY-SA) 4.0 International license but not necessarily in the public domain:  Other media outlets should be free to further disseminate the news while giving credit to the organization that produced it.   

Many countries have some form of [[w:community radio|community radio]].  Some of those radio stations include what they call news and / or public affairs, and some of those are made available as podcasts via the Internet.&lt;ref&gt;In the US, many of these stations collaborate via organizations such as the [[w:National Federation of Community Broadcasters|National Federation of Community Broadcasters]], the [[w:List of Pacifica Radio stations and affiliates#Radio Stations#Affiliates|Pacifica Network Affiliates]], and the [[w:Grassroots Radio Coalition|Grassroots Radio Coalition]]. One such station with regular local news produced by volunteers in [[w:KBOO|KBOO]] in [[w:Portland, Oregon|Portland, Oregon]]; see Loving (2019).&lt;/ref&gt; If their &quot;news &amp; public affairs&quot; programs are subsequently posted to a website as podcasts, preferably accompanied by some text if not complete transcripts, under a license no more restrictive than CC BY-SA, that should make them eligible for subsidies under the criteria mentioned above if they add at least one new podcast of that nature five days per week.  If the programming of this nature that they produce is ''not'' available on the web or under an appropriate license, part of any experiments as discussed here might include offers to help such radio stations become eligible.  

Might it be wise to allow children to vote for news organizations they like?  Ryan Sorrell, founder and publisher of the Kansas City Defender, insists that, &quot;young people ... are very interested in news. It just has to be produced and packaged the right way for them to be interested in consuming it&quot;.&lt;ref&gt;Holmes (2022).&lt;/ref&gt; The French-language [[:fr:w:Topo (revue)|''Topo'']] present news and complex issues in comic strip format. Their co-editor in chief insists, &quot;there are plenty of ways to get young people interested in current affairs&quot;.&lt;ref&gt;Biehlmann (2023).&lt;/ref&gt; Might allowing children to vote for news outlets they like increase public interest in learning and in civic engagement among both children and their caregivers?  Should this be tested in some experimental jurisdictions?&lt;ref&gt;We may not want infants who cannot read a simple children's book to vote for &quot;news&quot;, but if they can read the names of eligible local news outlets on a ballot, why not encourage them to vote?  As Rourmeen Islam wrote in 2002, &quot;erring on the side of more freedom rather than less would appear to cause less harm.&quot; (World Bank, 2002, pp. 21-22; 33-34/336 in pdf).&lt;/ref&gt;

Some of the money may go to media outlets that seem wacko to many voters.  However, how different might that be from the current situation? Most importantly, if these subsidies have the effect that Tocqueville reported from 1831, they should be good for democracy and for broadly shared peace and prosperity for the long term:  They could stimulate public debate, and wacko media might have ''less'' power than they currently do, with &quot;each separate journal exercis[ing] but little authority; but the power of the periodical press [being] second only to that of the people.&quot;&lt;ref&gt;Tocqueville (1835; 2001, p. 94).&lt;/ref&gt; 

Tocqueville's comparison of newspapers in France and the US in 1831 is echoed in Cagé's (2022) concern about &quot;the Fox News effect&quot; in the US and that of Bolloré in France.  She cites research claiming that biases in Fox News made major contributions to electing Republicans in the US since 2000.&lt;ref&gt;Cagé (2022, pp. 21-22, 59-60).  She cited DellaVigna and Ethan Kaplan (2007), who reported that Fox News had introduced cable programming into 20 percent of town in the US between 1996 and 2000.  They found that the presence of Fox increased the vote share for Republicans between 0.4 and 0.7 percentage points over neighboring non-Fox towns that seemed otherwise indistinguishable.  In 2000 Fox News was available in roughly 35 percent of households, which suggests that Fox News shifted the nationwide vote tally by between 0.15 and 0.2 percentage points.  They conclude that this shift was small but likely decisive in the close 2000 US presidential election.&lt;/ref&gt; These shifts, including changes by the conservative-leaning broadcasting company, Sinclair Broadcast Group, reportedly made a substantive contribution to the election of [[w:Donald Trump|Donald Trump]] as US President in 2016, while a comparable estimate of the impact of changes in MSNBC &quot;is an imprecise zero.&quot;&lt;ref&gt;Cagé (2022, pp. 21-22).  Miho (2022) analyzes the timing of the introduction of biased programming by the conservative-leaning broadcasting company, Sinclair Broadcast Group, between 1992 and 2020, comparing counties in the US with and without a Sinclair station.  This work estimates a 2.5 percentage point increase in the Republican vote share during the 2012 US presidential election and double that during the 2016 and 2020 presidential elections with comparable increases in Republican representation in the US Congress.&lt;/ref&gt; In France, she provides documentation claiming that the media empire of French billionaire [[w:Vincent Bolloré|Vincent Bolloré]] has made a major contribution to the rise of far-right politician [[w:Éric Zemmour|Éric Zemmour]] and is buying media in Spain.&lt;ref&gt;Cagé (2022, pp. 24, 60)&lt;/ref&gt; The pattern is simple:  Fire journalists and replace them with talk shows, which are cheaper to produce and are popular, evidently exploiting the [[w:overconfidence effect|overconfidence effect]].  

To what extent is the increase in political polarization since 1987&lt;ref name=fairness/&gt; and 2004&lt;ref&gt;Wikipedia &quot;[[w:Facebook|Facebook]]&quot;, accessed 2023-07-21.&lt;/ref&gt; due to increased concentration of ownership of both traditional and social media (and how those organizations make money selling changes in audience behaviors to the people who give them money)? To the extent that this increase in polarization has been driven by those changes in the media, citizen-directed subsidies for diverse news should reverse that trend. This hypothesis can be tested by experiments like those proposed herein.

== Roadmap for local news == 

Green et al. (2023) describe &quot;an emerging approach to meeting civic information needs&quot; in a &quot;Roadmap for local news&quot;. This report insists that society needs &quot;civic information&quot;, not merely &quot;news&quot;. It summarizes interviews with 51 leaders from nonprofit and commercial media across all forms of distribution (print, radio, broadcast, digital, SMS) in member organizations, news networks, news funders and researchers. They say that, &quot;Rampant disinformation is being weaponized by extremists&quot;, and &quot;Democratic participation and representation are under threat.&quot; They recommend four strategies to address &quot;this escalating information crisis&quot;: 

# Coordinate work around the goal of expanding “civic information,” not saving the news business;
# Directly invest in the production of civic information;
# Invest in shared services to sustain new and emerging civic information networks; and
# Cultivate and pass public policies that support the expansion of civic information while maintaining editorial independence. 

Part of the motivation for this article on &quot;Information is a public good&quot; is the belief that solid research on the value of such interventions should both (a) make it easier to get the funding needed, and (b) help direct the funding to interventions that seem to make the maximum contributions to improving broadly shared peace and prosperity for the long term at minimum cost.

== Budgets for experiments == 

What factors should be considered in evaluating budgets for experiments to estimate the impact of citizen-directed subsidies for news?   

[[File:Advertising as a percent of Gross Domestic Product in the United States.svg|thumb|Figure 9.  Advertising as a percent of Gross Domestic Product in the United States, 1919 to 2007.&lt;ref name=ads&gt;Galbi (2008).&lt;/ref&gt;]]  

Rolnik et al. (2019) suggested that $50 per person, roughly 0.08 percent of US GDP, might be enough.  However, that's a pittance compared to the revenue lost by newspapers in the US since 1955, as documented in Figure 8 above.  It's also a pittance compared to the money spent on advertising (see Figure 9):  Can we really expect local media funded with only 0.08 or 0.15 percent of GDP to compete with media funded by 2 percent of GDP?  Maybe, but that's far from obvious.  Might it be prudent to fund local journalism in some experimental jurisdictions at levels exceeding the money spent on advertising, i.e., at roughly 2 percent of GDP or more?  If information is a public good, as suggested by the research summarized here, then such high subsidies would be needed in some experimental jurisdictions, because the maximum of anything (including net benefits = benefits minus costs) cannot be confidently identified without conducting some experiments ''beyond the point of diminishing returns''.&lt;ref&gt;A parabola can be estimated from three distinct points.  However, in fitting a parabola or any other mathematical model to empirical data, one can never know if an empirical phenomenon has been adequately modeled and a maximum adequately estimated without data near the maximum and on both sides of it (unless the maximum is at a boundary, e.g., 0).  See, e.g., Box and Draper (2007).&lt;/ref&gt;  

[[File:AccountantsAuditorsUS.svg|thumb|Figure 10.  Accountants and auditors as a percent of the US workforce.&lt;ref name=actg&gt;Accountants and auditors as a percent of US households, 1850 - 2016, using the OCC1950 occupation codes in a sample of households available from from the [[w:IPUMS|Integrated Public Use Microdata Series at the University of Minnesota (IPUMS)]].  For more detail see the &quot;AccountantsAuditorsPct&quot; data set in the &quot;Ecdat&quot; package and the &quot;AccountantsAuditorsPct&quot; vignette in the &quot;Ecfun&quot; package available from within the [[w:R (programming language)|R (programming language)]] using 'install.packages(c(&quot;Ecdat&quot;, &quot;Ecfun&quot;), repos=&quot;http://R-Forge.R-project.org&quot;)'.&lt;/ref&gt;]] 

Also, news might serve a roughly comparable function to accounting and auditing, as both help reduce losses due to incompetence, malfeasance and fraud.  Two points on this:  

# CONTROL FRAUDS:  Black (2013) noted that many heads of organizations can find accountants and auditors willing to certify accounting reports they know to be fraudulent. Black calls such executives &quot;control frauds.&quot;&lt;ref&gt;Black (2013). &lt;/ref&gt; Primary protections against these kinds of problems are vigorous, independent journalists and more money spent on independent evaluations beyond the control of such executives.  In this regard, we note two major differences between the [[w:Savings and loan crisis|Savings &amp; Loan scandal]] of the late 1980s and early 1990s&lt;ref&gt;Wikipedia &quot;[[w:Savings and loan crisis|Savings and loan crisis]]&quot;, accessed 2023-06-25.&lt;/ref&gt; and the [[w:2007–008 financial crisis|international financial meltdown of 2007-2008]]:&lt;ref&gt;Wikipedia &quot;[[w:2007–008 financial crisis|2007–008 financial crisis]]&quot;, accessed 2023-06-25.&lt;/ref&gt; First the major banks by 2007 were much bigger and controlled much larger advertising budgets than the Saving &amp; Loan industry did 15-20 years earlier.  This meant that major media had a much bigger conflict of interest in honestly reporting on questionable activities of these major accounts.  Second, the major banks had made much larger political campaign contributions to much larger portions of both the US House and Senate. However, might the massive amounts of big money spent on campaign finance have been as effective if the major media did not have such a conflict of interest in exposing more details of the corrosive impact of major campaign donors on the quality of government?  To what extent might this corrosive impact be quantified in experimental polities?  
# ADEQUATE RESEARCH OF OUTCOMES:  Many nonprofits and governmental agencies officially have outcome measures, but many of those measures tend to be relatively superficial like the number of people served.  It's much harder to evaluate the actual benefits to the people served and to society.  For example, the Perry Preschool&lt;ref&gt;Schweinhart et al. (2005).  See also Wikipedia &quot;[[w:HighScope|HighScope]]&quot;, accessed 2023-06-15. &lt;/ref&gt; and Abecedarian&lt;ref&gt;e.g., Sparling and Meunier (2019).  See also Wikipedia &quot;Abecedarian Early Intervention Project&quot;, accessed 2023-06-25.&lt;/ref&gt; programs divided poor children and caregivers into experimental and control groups and followed them for decades to establish that their interventions were enormously effective.&lt;ref&gt;For more recent research on the economic value of high quality programs for early childhood development, see, e.g., &lt;!-- &quot;The Heckman Equation&quot; website (heckmanequation.org)--&gt;{{cite Q|Q121010808}}, accessed 2023-07-29.&lt;/ref&gt; Meanwhile, US President Lyndon Johnson's [[w:Great Society|Great Society]] programs,&lt;ref&gt;Wikipedia &quot;[[w:Great Society|Great Society]]&quot;, accessed 2023-07-11.&lt;/ref&gt; and [[w:Head Start|Head Start]] in particular, did not invest as heavily in research.  That lack of documentation of results made them a relatively easy target for political opponents claiming that government is the problem, not the solution. These counter arguments were popularized by US President Ronald Reagan and UK Prime Minister Margaret Thatcher to justify reducing or eliminating government funds for many such programs. Banerjee and Duflo (2019) summarized relevant research in this area by saying that the programs were not the disaster that Reagan, Thatcher, and others claimed, but they were also not as efficient and effective as they could have been, because many local implementations were underfunded, poorly managed and poorly evaluated.   Bedasso (2021) analyzed World Bank projects completed from 2009 to 2020, concluding that high quality monitoring and evaluation on average made a major contribution to the positive results from the successful projects studied.&lt;ref&gt;See also Raimondo (2016).&lt;/ref&gt; To what extent might citizen-directed subsidies for local media as suggested here improve the demand for (and the supply of) better evaluations, leading to better programs and the lower crime, etc., that came from those programs?  To what extent might this effect be quantified using randomized controlled trials comparing different jurisdictions, analogous to the research for which Banerjee, Duflo, and Kramer won the 2019 Nobel Memorial Prize in Economics?  

This discussion makes us wonder if better research and better news might deliver dramatically more benefits than costs in reducing money wasted on both funding wasteful programs and on failing to fund effective ones?  In particular, might society benefit from matching the 1 percent of the workforce occupied by accountants and auditors with better research and citizen-directed subsidies for news (see Figure 10)?  If, for example, 1 or 2 percent of GDP distributed to local news nonprofits via local elections, as described above, increased the average rate of economic growth in GDP per capita by 0.1 percentage point per year, that increase would accumulate over time, so that after 10 or 20 years, the news would in effect become free, paid by money that implementing political jurisdictions would not have without those subsidies. Moreover those accumulations might remain as long as they were not wiped out by events comparable to the economic disasters documented above in discussing &quot;Stalin and Putin&quot; -- and maybe not even then as suggested by Figure 7. 

=== Other recommendations and natural experiments === 

Table 1 compares the recommendations of McChesney and Nichols (2021, 2022) and Rolnik et al. (2019) with other possible points of reference. Crudely similar to McChesney, Nichols, and Rolnik et al., Karr (2019) and Karr and Aaron (2019) recommend &quot;a 2 percent ad tax on all online enterprises that in 2018 earned more than $200 million in annual digital-ad revenues&quot;. They claim that this &quot;would yield more than $1.8 billion a year&quot;,  which is very roughly 0.008 percent of GDP, $5 per person per year;&lt;ref name=Karr&gt;Karr (2019), Karr and Aaron (2019). US GDP for 2019 was $21,381 billion, per International Monetary Fund (2023). Thus, $1.8 billion is 0.0084% of US GDP and $5.44 for each of the 330,513,000 humans in the US in 2019; round to 0.008% and $5 per capita.&lt;/ref&gt; Google has negotiated agreements similar to this with the governments of Australia and Canada.&lt;ref&gt;Hermida (2023).&lt;/ref&gt; Other points of reference include the percent of GDP devoted to accounting and auditing and advertising. As displayed in Figure 10, accountants and auditors are roughly 1 percent of the workforce in the US. It's not clear how to translate that into a percent of GDP, but 2 percent seems like a reasonable approximation, if we assume that average income of accountants and auditors is a little above the national norm and overhead is not quite double their salaries; this may be conservative, because many accountants and auditors have support staff, who are not accountants nor auditors but support their work. 

Another point of reference is the average annual growth rate in GDP per capita since World War II: A subsidy of 2 percent of GDP would be roughly one year's increase in average annual income since World War II, as noted with Figure 1 above. More precisely, the US economy (GDP per capita adjusted for inflation) was 2.2 percent per year between between 1950 and 1990 but only 1.1 percent between 1990 and 2022, according to inequality expert [[w:Thomas Piketty|Thomas Piketty]], who attributed that slowing in the rate of economic growth to the increase in income inequality in the US since 1975, documented in Fgure 6 above. Whether Piketty is correct or not, if 2 percent per year subsidies for journalism close the gap between 1.1 and 2.2 percent per year, those media subsidies would effectively become free after two years, paid out of income the US would not have without them. This reinforces the main point of this essay regarding the need for randomized controlled trials on any intervention with a credible claim to improving the prospects for broadly shared economic growth for the long term.&lt;ref name=GDP&gt;The growth in US GDP per capita is discussed in the working paper on Wikiversity titled, &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;, accessed 2021-05-19.   For a similar comment about an intervention that increased the rate of economic growth becoming free, paid out of income we would not have without it, has been made about the impact of improving education by {{cite Q|Q56849246}}&lt;!-- Endangering Prosperity: A Global View of the American School--&gt;, p. 12.&lt;/ref&gt;    
This table includes other interventions for which humanity would benefit from more substantive evaluation of their impact. This includes [[w:Democracy voucher|Seattle's &quot;Democracy Voucher&quot; program]], which gives each registered voter four $25 vouchers, totaling $100, which they could give to eligible candidates running for municipal office. However, only the first 47,000 were honored;  this limited the city's commitment to $4.7 million every other year.&lt;ref name=Berman&gt;Berman (2015). The Wikipedia article on [[w:Seattle]] says that the gross metropolitan product (GMP) for the Seattle-Tacoma metropolitan area was $231 billion in 2010 for a population of 3,979,845.  That makes the GMP per capita roughly $58,000.  However, the population of Seattle proper was only 608,660 in 2010, making the Gross City Product roughly $35 billion.  $4.7 million is 0.0133 percent of $35 billion.  However, that's very other year, so it's really only 0.007 percent of the Gross City Product.&lt;/ref&gt; If Seattle can afford $100 per registered voter, many other governmental entities can afford something very roughly comparable for each adult in their jurisdiction. Seattle's &quot;democracy vouchers&quot; are used to fund political campaigns, not local media; they are mentioned here as a point of comparison. 

Other interventions that seem to deserve more research than we've seen are the [[w:New Jersey Civic Information Consortium|New Jersey Civic Information Consortium]] (NJCIC)&lt;ref name=njcicBudget&gt;Karr (2020). Per [[w:New Jersey]] the Gross State Product in 2018 was roughly $640 billion;  it's population in 2020 was roughly 9.3 million. The initial  $500,000 for the project is only $0.05 per person per year and only 0.00008 percent of $640 billion.&lt;/ref&gt; and a program in California to improve local news in communities in dire need of strong local journalism. The NJCIC was initially funded at $500,000, which is only 0.00008 percent of New Jersey economy (GDP) of $640 billion. In 2022, the state of California authorized $25 million for up to 40 Berkeley local news fellowships offering &quot;a $50,000 annual stipend [for 3 years] to supplement their salaries while they work in California newsrooms covering communities in dire need of strong local journalism.&quot; This Berkeley program is roughly $0.21 per person per year, 0.0007 percent of the Gross State Produce of $3.6 trillion that year, for an annual rate of very roughly 0.0002 percent of the Gross State Product.&lt;ref name=Berkeley&gt;Natividad (2023) discusses the Berkeley local news fellowships. California Gross State Product from US Bureau of Economic Analysis (2023). California population on 2022-07-01 from US Census Bureau (2023).&lt;/ref&gt; A similar project in Indiana funded by philanthropies began as the Indiana Local News Initiative&lt;ref&gt;Greenwell (2023).&lt;/ref&gt; and has morphed into Free Press Indiana.&lt;ref&gt;See &quot;[https://www.localnewsforindiana.org LocalNewsForIndiana.org]&quot;; accessed 29 December 2023.&lt;/ref&gt;

Some local [[w:League of Women Voters|Leagues of Women Voters]] have all-volunteer teams who observe official meetings of local governmental bodies and write reports.&lt;ref&gt;Wilson (2007).&lt;/ref&gt; The [[w:City Bureau|City Bureau]] nonprofit news organization in Chicago, Illinous, &quot;trains and pays community members to attend local government meetings and report back on them.&quot;&lt;ref&gt;See &quot;[https://www.citybureau.org/documenters-about citybureau.org/documenters-about]&quot;.&lt;/ref&gt; The program has been so successful, it has exanded to other cities.&lt;ref&gt;Greenwell (2023).&lt;/ref&gt;

For an international comparison, we include [[w:amaBhungane|amaBhungane]],&lt;ref name=amaBhu&gt;The budget for [[w:AmaBhungane#Budget|amaBhungane]] in 2020 was estimated at 590,000 US dollars at the current exchange rate, per analysis in the [[w:AmaBhungane#Budget|budget]] section of the Wikipedia article on amaBhungane. That's 0.00017 percent of South African's nominal GDP for that year of 337.5 million US dollars, per the section on &quot;[[w:Economy of South Africa#Historical statistics 1980–2022|Historical statistics 1980–2022]]&quot; in the Wikipedia article on [[w:Economy of South Africa|Economy of South Africa]]; round that to 0.002 percent for convenience. The population of South Africa that year was estimated at 59,309,000, according to the section on &quot;[[w:Demographics of South Africa#UN Age and population estimates: 1950 to 2030|UN Age and population estimates: 1950 to 2030]]&quot; in the Wikipedia article on [[w:Demographics of South Africa|Demographics of South Africa]]; this gives a budget of 1 penny US per capita. (All these Wikipedia articles were accessed 2023-12-28.)&lt;/ref&gt; whose investigative journalism exposed a corruption scandal that helped force South African President [[w:Jacob Zuma|Jacob Zuma]] to resign in 2018; amaBhungane's budget is very roughly one penny US per person per year in South Africa, 0.0002 percent of GDP. To the extent that this essay provides a fair and balanced account of the impact of journalism on political economy, South African and the rest of the world would benefit from more funding for amaBhungane and other comparable investigative journalism organizations. This could initially include randomized controlled trials involving citizen-directed subsidies for local news outlets in poor communities in South Africa and elsewhere, as we discuss further in the rest of this essay. Without such experiments, we are asking for funds based more on faith than science. 

{| class=&quot;wikitable sortable&quot; 
!option / reference
!% of GDP 
!colspan=2|US$
!per …
!
|-
|style=&quot;text-align:left;&quot;|US postal subsidies for newspapers 1840-44 
| 0.21%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $140
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
| &lt;ref name=McC-N2010/&gt;
|-
|style=&quot;text-align:left;&quot;|McChesney &amp; Nichols (2021, 2022)
| 0.15%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $100
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
| &lt;ref name=McC-N2021/&gt;
|-
|[[Confirmation bias and conflict#Relevant research|Rolnik et al.]] 
| 0.08%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $50
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|adult &amp; year
| &lt;ref name=Rolnik/&gt; 
|-
|[[w:Free Press (organization)|Free Press]]
| 0.008%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $6
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref name=Karr/&gt;
|-
|[[w:Democracy vouchers|Democracy vouchers]]
| 0.007% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $100
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|voter &amp; municipal election for the first 47,000
|&lt;ref name=Berman/&gt;
|-
|[[Confirmation bias and conflict#Advertising and accounting|advertising]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref name=ads/&gt;
|-
|[[Confirmation bias and conflict#Advertising and accounting|accounting]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref&gt;As noted with Figure 10 and the discussion above, accountants and auditors are roughly 1 percent of the US workforce, and it seems reasonable to guess that their pay combined with support staff and overhead would likely make them roughly double that, 2 percent, as a portion of GDP.&lt;/ref&gt;
|-
|[[US Gross Domestic Product (GDP) per capita|US productivity improvements]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year (GDP per capita)
|&lt;ref&gt;For an analysis of the rate of growth in US GDP per capita, see the working paper on Wikiversity titled, &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;&lt;/ref&gt;
|-
|[[w:New Jersey Civic Information Consortium|New Jersey Civic Information Consortium]]
| 0.00008%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .05
|person &amp; year
|&lt;ref name=njcicBudget/&gt;
|-
| Berkeley local news fellowships  
| 0.0002%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .21
|person &amp; year
|&lt;ref name=Berkeley/&gt;
|-
|[[w:amaBhungane|amaBhungane]]
| 0.0002%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .01
|person &amp; year in South Africa
|&lt;ref name=amaBhu/&gt;
|}
Table 1. Media subsidies and other points of reference. At the low end, political corruption exposed in part by amaBhungane forced the resignation in 2018 of South African President Zuma on a budget that's very roughly one penny US per person per year. If much higher subsidies of 1 percent of GDP restored an annual growth rate of 2.2 percent per year from the more recent 1.1 percent discussed with Figure 1 above, those subsidies would pay for themselves from one year's growth that the US would not otherwise have.

== Other factors == 

We feel a need here to suggest other issues to consider in designing experiments to improve the political economy:  education, empowering women, free speech, free press, peaceful assembly, and reducing political polarization.

EDUCATION:  Modern research suggests that society might have lower crime&lt;ref&gt;Wang et al. (2022).&lt;/ref&gt; and faster rates of economic growth with better funding for and better research&lt;ref&gt;Hanushek and Woessmann (2015).&lt;/ref&gt; on quality child care from pregnancy through age 17. 

EMPOWERING WOMEN:  Might the best known way to limit and reverse population growth be to empower women and girls?  Without that, might the human population continue to grow until some major disaster reduces that population dramatically?&lt;ref&gt;Roser (2017).&lt;/ref&gt;

FREE SPEECH, FREE PRESS, PEACEFUL ASSEMBLY:  Verbitsky said, &quot;Journalism is disseminating information that someone does not want known; the rest is propaganda.&quot;&lt;ref&gt;Verbitsky (2006, p. 16), author's translation from Spanish.&lt;/ref&gt; This discussion of threats, arrests, kidnappings, and murders of journalists&lt;ref&gt;Monitored especially by the [[w:Committee to Protect Journalists|Committee to Protect Journalists]], as discussed in the Wikipedia article on them, accessed 2023-07-04.&lt;/ref&gt; and violent suppression of peaceful assemblies&lt;ref&gt;Monitored by Freedom House and others.  See, e.g., the Wikipedia article on &quot;[[w:List of freedom indices|List of freedom indices]]&quot;, accessed 2023-07-04.&lt;/ref&gt; encourages us to consider the potential utility of efforts to improve local news, as noted by contributors to Islam et al., eds. (2002), cited above.  Data on such problems should be considered in selecting sites for experiments with citizen-directed subsidies for journalism and in analyzing the results from such experiments. Such data should include the incidence of legal proceedings against journalists and publishers&lt;ref&gt;Including the risks of [[w:trategic lawsuit against public participation|strategic lawsuits against public participation]] (SLAPPs) and other questionable uses of the courts including some documented in the &quot;[[w:Freedom of the Press Foundation#U.S. Press Freedom Tracker|U.S. Press Freedom Tracker]]&quot;, mentioned above.&lt;/ref&gt; as well as threats, murders, etc., in jurisdictions comparable to experimental jurisdictions.  

REDUCING POLITICAL POLARIZATION:  What interventions might be tested that would attempt to reduce political polarization while also experimenting with increasing funding for news through small, diverse news organizations? For example, might an ad campaign feature someone saying, &quot;We don't talk politics&quot;, with a  reply, &quot;We have to talk politics with humility and mutual respect, because the alternative is killing people over misunderstandings&quot;? Might another ad say, &quot;Don't get angry: Get curious&quot;?  What can be done to encourage people to get curious rather than angry when they hear something that contradicts their preconceptions?  How can people be encouraged to talk politics with humility and respect for others, understanding that everyone can be misinformed and others might have useful information?&lt;ref&gt;Wikiversity &quot;[[How can we know?]]&quot;, accessed 2023-07-22, reviews relevant research relating to political polarization.  Yom-Tov et al. (2018) described a randomized-controlled trial that compared the effectiveness of different advertisements &quot;to improve food choices and integrate exercise into daily activities of internet users.&quot; They found &quot;powerful ways to measure and improve the effectiveness of online public health interventions&quot; and showed &quot;that corporations that use these sophisticated tools to promote unhealthy products can potentially be outbid and outmaneuvered.&quot; Similar research might attempt to promote strategies for countering political polarization.  See also Piwowarski  et al. (2019).&lt;/ref&gt;  

FOCUS ON POLITICIANS: Mansuri et al. (2023) randomly assigned presidents of village governments in the state of [[w:Tamil Nadu|Tamil Nadu]] in India to one of three groups with (1) a financial incentive or (2) a certificate with an information campaign (without a financial incentive) for better government or (3) a control group. They found that the public benefitted from both the financial and non-financial incentives, and the non-financial incentives were more cost effective. Might it make sense in some experimental jurisdictions to structure the subsidies for local news by asking voters to allocate, e.g., half their votes for local news to outlet(s) that they think provide the best information about politicians with the other half based on &quot;general news&quot;?&lt;ref&gt;Mansuri et al. (2023).&lt;/ref&gt;

PIGGYBACK ON COMMUNITY AND LOCAL DEVELOPMENT PROGRAMS:  The World Bank (2023) notes that, &quot;Experience has shown that when given clear and transparent rules, access to information, and appropriate technical and financial support, communities can effectively organize to identify community priorities and address local development challenges by working in partnership with local governments and other institutions to build small-scale infrastructure, deliver basic services and enhance livelihoods. The World Bank recognizes that CLD [Community and Local Development]  approaches and actions are important elements of an effective poverty-reduction and sustainable development strategy.&quot; This suggests that experiments in citizen-directed subsidies for news might best be implemented as adjuncts to other CLD projects to improve &quot;access to information&quot; needed for the success of that and follow-on projects.  Such news subsidies should complement and reinforce the quality of monitoring and evaluation, which was &quot;significantly and positively associated with project outcome as institutionally measured at the World Bank&quot;.&lt;ref&gt;Raimondo (2016). See also Bedasso (2021).&lt;/ref&gt;

Simple experiments of the type suggested here might add citizen-directed subsidies for local news to a random selection of Community and Local Development (CLD) projects.  Such experimental jurisdictions should be small enough that the budget for the proposed citizen-directed subsidies for news would be seen as feasible but large enough so appropriate data could be obtained and compared with control jurisdictions not receiving such subsidies for local news.

However, some potential recipients of CLD funding may be in news deserts or with &quot;ghost newspapers&quot;, as mentioned above.  Some may not have at least three local news outlets that have been publishing something they call news each workday for at least six months, as required for the local elections recommended by McChesney and Nichols (2021, 2022), outlined above.  In such jurisdictions, the local consultations that identify community priorities for CLD funding should also include discussions of how to grow competitive local news outlets to help the community maximize the benefits they get from the project.  

The need for &quot;at least three local news outlets&quot; is reinforced by the possibilities that two or three local news outlets may be an [[w:oligopoly|oligopoly]], acting like a monopoly.  This risk may be minimized by working to ''limit'' barriers to entry and to encourage different news outlets to serve different segments of the market for news.  The risks of oligopolistic behavior may be further reduced by requiring all recipients of citizen-directed subsidies to release their content under a free license like the Creative Commons Attribution-ShareAlike (SS BY-SA) 4.0 international license.  This could push each independent local news outlet to spend part of their time reading each other's work while pursuing their own journalistic investigations, hoping for scoops that could attract a wider audience after being cited by other outlets.&lt;ref&gt;Wikipedia &quot;[[w:Oligopoly|Oligopoly]]&quot;, accessed 2023-07-06.&lt;/ref&gt;

This preference for at least three independent local news outlets in an experimental jurisdiction puts a lower bound on the size of jurisdictions to be included as experimental units, especially if we assume that the independent outlets should employ on average at least two journalists, giving a minimum of six journalists employed by local news outlets in an experimental jurisdiction.  The discussions above suggested subsidies ranging from 0.08 percent to 2 percent or more.  To get a lower bound for the size of experimental jurisdictions, we divide 6 by 0.08 and 2 percent:  Six journalists would be 0.08 percent of a population of 7,500 and 2 percent of a population of 300.

== Sampling units / experimental polities == 

Many local governments could fund local news nonprofits at 0.15% of GDP, because it would likely be comparable to what they currently spend on accounting, media and public relations.&lt;ref&gt;&quot;State and local governments [in the US] spent $3.5 trillion on direct general government expenditures in fiscal year 2020&quot;, with states spending $1.7 trillion and local governments $1.8 trillion&quot;, per Urban Institute (2024). The nominal GDP of the US for 2020 was $21 trillion, per International Monetary Fund (2024). Thus, local government spending $1.7 trillion is 1.8% of the $21 trillion US GDP, which is comparable to the money spent on accounting per Figure 10 and advertising per Figure 9.&lt;/ref&gt; If the results of such funding are even a modest percent of the benefits claimed in the documents cited above, any jurisdiction that does that would likely obtain a handsome return on that investment. 

Jurisdictions for randomized controlled trials might include some of the members of the United Nations with the smallest Gross Domestic Products (GDPs) or even some of the poorest census-designated places&lt;ref&gt;Wikipedia &quot;[[w:Census-designated place|Census-designated place]]&quot;, accessed 2023-07-11.&lt;/ref&gt; in a country like the US. Alternatively, they might include areas with seemingly intractable cycles of violence like Israel and Palestine: The budget for interventions like those proposed herein are a fraction of what is being spent on defense and on violence challenging existing power structures. If interventions roughly comparable to those discussed herein can reduce the lethality of a conflict at a modest cost, it would have an incredible return on investment (ROI). That would be true not merely for the focus of the intervention but for other similar conflicts. 

For illustration purposes only, Table 2 lists the six countries in the United Nations with the smallest GDPs in 2021 in US dollars at current prices according to the United Nations Statistics Division plus Palestine and Israel, along with their populations and GDP per capita plus the money required to fund citizen-directed subsidies at 0.15 percent of GDP, as recommended by McChesney and Nichols (2021, 2022).  The rough budgets suggested here would only be for a news subsidy companion to Community and Local Development (CLD) projects, as discussed above or for intervention(s) attempting to reduce the lethality of conflict.  Other factors should be considered in detailed planning.  For example, the budget for such a project in [[w:Montserrat|Montserrat]] may need to be increased to support greater diversity in the local news outlets actually subsidized.  And a careful study of local culture in [[w:Kiribati|Kiribati]] may indicate that the suggested budget figure there may support substantially fewer than the 97 journalists suggested by the naive computations in this table.  The key point, however, is that subsidies of this magnitude would be modest as a proportion of many other projects funded by agencies like the World Bank or the money spent on defense or war. 

{| class=&quot;wikitable sortable&quot; style=text-align:right
! Country !! Population !! GDP / capita
! GDP (million USD) 
! annual subsidy at 0.15% of GDP ($K) 
! number of journalists&lt;sup&gt;(*)&lt;/sup&gt;
|-
| [[w:Tuvalu|Tuvalu]] || 11,204 || $5,370 || $60 || $90 ||8.4
|-
| [[w:Montserrat|Montserrat]] || 4,417 || $16,199 || $72 || $107 || 3.3
|- 
| [[w:Nauru|Nauru]] || 12,511 || $12,390 || $155 || $233 || 9.4
|- 
| [[w:Palau|Palau]] || 18,024 || $12,084 || $218 || $327 || 13.5
|- 
| [[w:Kiribati|Kiribati]] || 128,874 || $1,765 || $227 || $341 ||96.7
|- 
| [[w:Marshall Islands|Marshall Islands]] || 42,040 || $6,111 || $257 || $385 || 31.5
|- 
| [[w:State of Palestine|State of Palestine]] || 5,483,450 || $3,302 || $18,037 || $27,055 || 4,113
|- 
| [[w:Israel| Israel]] || 9,877,280 || $48,757 || $481,591 || $722,387 || 7,408 
|}
Table 2.  Rough estimate of the budget for subsidies at 0.15 percent of GDP for the 6 smallest members of the UN plus Palestine and Israel. Population and GDP at current prices per United Nations Statistics Division (2023). (*) &quot;Number of journalists&quot; was computed assuming each journalist would cost twice the GDP / capita.  For example, the GDP / capita for Tuvalu in this table is $5,370. Double that to get $10,740.  Divide that into $90,000 to get 8.4.  

Other possibilities for experimental units might be historically impoverished subnational groups like [[w:Native Americans in the United States|Native American jurisdictions in the United States]].  As of 12 January 2023 there were &quot;574 Tribal entities recognized by and eligible for funding and services from the [[w:Bureau of Indian Affairs|Bureau of Indian Affairs]] (BIA)&quot;, some of which have multiple subunits, e.g., populations in different counties or census-designated places. For example, the largest is the [[w:Navajo Nation|Navajo Nation Reservation]] that is split between Arizona, New Mexico, and Utah.&lt;ref&gt;Newland (2023).&lt;/ref&gt; Some of these subdivisions are too small to be suitable for experiments in citizen-directed subsidies for news.  Others have subdivisions large enough so that some subdivisions might be in experimental group(s) with others as controls.  If there are at least three subdivisions with sufficient populations, at least one could be a control, with others being Community and Local Development (CLD) projects both with and without companion news subsidies, as discussed above.&lt;ref&gt;Data analysis might consider spatial autocorrelation, as used by Mohammadi et al. (2022) and multi-level time series text analysis, used by Friedland et al. (2022).  The latter discuss &quot;Asymmetric communication ecologies and the erosion of civil society in Wisconsin&quot;:  That state had historically been moderate &quot;with a strong progressive legacy&quot;.  Then in 2010 they elected a governor who attacked the state's public sector unions with substantial success and voted for Donald Trump for President in 2016 but ''against'' him in 2020.&lt;/ref&gt;

== Supplement not replace other funding == 

The subsidies proposed here should supplement not replace other funding, similar to the subsidies under the US Postal Service Act of 1792.  McChesney and Nichols recommended that an organization should be publishing something they call news five days per week for at least six months, so the voters would know what they are voting for.  Those criteria might be modified, at least in some experimental jurisdictions, especially in news deserts, as something else is done to create local news organizations eligible to receive a portion of the experimental citizen-directed subsidies.  

The [[w:Institute for Nonprofit News|Institute for Nonprofit News]] and Local Independent Online News (LION) Publishers&lt;ref&gt;&lt;!-- Local Independent Online News (LION) Publishers--&gt;{{cite Q|Q104172660}}&lt;/ref&gt; help local news organizations get started and maintain themselves.  Organizations like them might help new local news initiatives in experimental jurisdictions as discussed in this article.

== Funding research in the value of local news ==

We would expect two sources of funding for research to quantify the value of local news 

* The [[w:World Bank|World Bank]] has already discussed the value of news. We would expect that organizations that fund community and local development projects would also want to fund experiments in anything that seemed likely to increase the return on their investments in such projects. 
* Folkenflik (2023) wrote, &quot;Some of the biggest names in American philanthropy have joined forces to spend at least $500 million over five years to revitalize the coverage of local news in places where it has waned.&quot; This group of philanthropic organizations includes the American Journalism Project, which says they &quot;measure the impact of our philanthropic investments and venture support by evaluating our efficacy in catalyzing grantees’ organizational growth, sustainability and impact.&quot;&lt;ref&gt;Website of the American Journalism Project accessed 29 December 2023 ([https://www.theajp.org/about/impact/# https://www.theajp.org/about/impact/#]).&lt;/ref&gt; 

If the claims made above for the value of news are fair, then appropriate experiments might be able to quantify who benefit from improving the news ecology, how much they benefit, which structures seem to work the best, and even the optimal level of funding.

== Summary == 

This article has summarized numerous claims regarding different ways in which information is a public good.  Many such claims can be tested in experiments crudely similar to those for which Banerjee, Duflo, and Kremer won the [[w:2019 Nobel Memorial Prize in Economic Sciences|2019 Nobel Memorial Prize in Economics]].  We suggest funding such projects as companions to Community and Local Development (CLD) projects.  If the research cited above is replicable, the returns on such investments could be huge, delivering benefits to the end of human civilization, similar to those claimed for newspapers published in the US in the early nineteenth century, which may have made major contributions to carrying the US to its current position of world leadership and to developing technologies that benefit the vast majority of humanity the world over today.

== Acknowledgements == 

Thanks especially to Bruce Preville who pushed for evidence supporting wide ranging claims of media influence in limiting progress against many societal ills.  Thanks also to Dave Black for suggesting experimenting with Native American jurisdictions in the US and to Joy Ellsworth for describing the substantial cultural challenges that such interventions might face.

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* &lt;!-- Leigh Thompson and George Loewenstein (1992) &quot;Egocentric interpretations of fairness and interpersonal conflict&quot;, Organizational Behavior and Human Decision Processes. 51 (2): 176–197--&gt;{{cite Q|Q29300051}}
* &lt;!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) --&gt;{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}}
* &lt;!-- Rachel Treisman (2021)  When this hedge fund buys local newspapers, democracy suffers--&gt;{{cite Q|Q121438095}}
* &lt;!-- United Nations (2022) World Population Prospects 2022, Online Edition (Department of Economic and Social Affairs, Population Division) --&gt;{{cite Q|Q113015862|author=United Nations|publisher=unset}}
* &lt;!-- United Nations Statistics Division (2021) &quot;Basic Data Selection&quot; --&gt;{{cite Q|Q120485981|author=United Nations Statistics Division|date=2023-07-06|publisher=unset}}
* &lt;!-- US Bureau of Economic Analysis (2023-03-31) Gross Domestic Product by State, 4th Quarter and Annual 2022--&gt;{{cite Q|Q124036982|author=US Bureau of Economic Analysis}}
* &lt;!-- US Census Bureau (2023) Quick facts--&gt;{{cite Q|Q124037268|author=US Census Bureau}}
* &lt;!--US Institute of Peace (2016) Tools for Improving Media Interventions in Conflict Zones--&gt;{{cite Q|Q124692870|author =US Institute of Peace}}
* &lt;!-- Siva Vaidhyanathan (2018) Antisocial Media: How Facebook Disconnects Us and Undermines Democracy (Oxford U. Pr.) --&gt;{{cite Q|Q56027099}}
* &lt;!-- Alexandra Wake, Abbas Valadkhani, Alan Nguyen, and Jeremy Nguyen (2021-04-15) &quot;The Economic Costs of Curbing Press Freedom&quot;, Global Investigative Journalism Network--&gt;{{cite Q|Q120739929}}
* &lt;!-- Leah Wang, Wendy Sawyer, Tiana Herring and Emily Widra (2022) &quot;Beyond the count: A deep dive into state prison populations&quot;, Prison Policy Initiative --&gt;{{cite Q|Q120332620}}
* &lt;!-- Garrett Watson (2022) Combined Federal and State Corporate Income Tax Rates in 2022 --&gt;{{cite Q|Q120793929}}  
* &lt;!-- Richard G. Wilkinson and Kate E. Pickett (2017) &quot;The Enemy between Us: The Psychological and Social Costs Of Inequality&quot;, European Journal of Social Psychology, 47(1)11-24 --&gt;{{cite Q|Q118385024}}
* &lt;!--Mary G. Wilson (2007) Observing Your Government in Action: Protecting your right to know (LWV)--&gt;{{cite Q|Q124048761}}
* &lt;!-- World Bank (2022) Annual Report --&gt;{{cite Q|Q118270948|author=World Bank|series=unset}}
* &lt;!-- World Bank (2023) &quot;Community and Local Development&quot; (web page:  &quot;https://www.worldbank.org/en/topic/communitydrivendevelopment&quot;) --&gt;{{cite Q|Q120380457|author=World Bank|date=2023-03-27|publisher=unset}}
* &lt;!-- Elad Yom-Tov, Jinia Shembekar, Sarah Barclay, and Peter Muennig (2018) &quot;The effectiveness of public health advertisements to promote health: a randomized-controlled trial on 794,000 participants&quot;--&gt;{{cite Q|Q91869157}}
* &lt;!-- Erica York (2023-01-26) &quot;Summary of the Latest Federal Income Tax Data, 2023 Update&quot;, Tax Foundation --&gt;{{cite Q|Q118189145}}
* &lt;!-- Shoshana Zuboff (2019) The Age of Surveillance Capitalism (PublicAffairs) --&gt;{{cite Q|Q75804726}}
* James D. Wolfensohn (2002) Foreword to Islam et al. (2002, pp. v-vi).

== Notes ==
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:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].''  

::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' 

== Abstract == 

This article reviews literature relevant to the claim that &quot;information is a public good&quot; and recommends experiments to quantify the impact of news on society, including on violent conflicts and broadly shared economic growth. We propose randomized controlled trials to evaluate the relative effectiveness of alternative interventions on the lethality of conflict and broadly shared economic growth. Experimental units would be polities in conflict or with incomes (nominal Gross Domestic Products, GDPs or gross local products) small enough so competitive local news outlets could be funded by philanthropies or organizations like the World Bank but large enough that their political economies have been tracked with sufficient accuracy to allow them to be considered in such an experiment. One factor in such experiments would be subsidies for local journalism, perhaps distributed to local news outlets on the basis of local elections, similar to the proposal of McChesney and Nichols (2021, 2022).

== Introduction ==

:''Information is a public good.''&lt;ref&gt;This is the title of Cagé and Huet (2021, in French).  However, the thrust of their book is very different.  It is subtitled, &quot;Refounding media ownership&quot;.  Their focus is on creating legal structure(s) to support journalistic independence as outlined in Cagé (2016).&lt;/ref&gt; 
:''Misinformation is a public nuisance.''&lt;ref&gt;The Wikipedia article on &quot;[[w:Public nuisance|Public nuisance]]&quot; says, &quot;In English criminal law, public nuisance was a common law offence in which the injury, loss, or damage is suffered by the public, in general, rather than an individual, in particular.&quot; (accessed 2023-04-24.)&lt;/ref&gt;
:''Disinformation is a public evil.''&lt;ref&gt;The initial author of this essay is unaware of any previous use of the term, &quot;public evil&quot;, but it seems appropriate in this context to describe content disseminated by mass media, including social media, curated with the explicit intent to convince people to support public policies contrary to the best interests of the audience and the general public.&lt;/ref&gt; 

=== Public goods ===

In economics, a [[w:public good|public good]] is a good (or service) that is both [[w:non-rivalrous|non-rivalrous]] and [[w:non-excludable|non-excludable]].&lt;ref&gt;e.g., Cornes and Sandler (1996).&lt;/ref&gt; Non-rivalrous means that we can all consume it at the same time. An apple is rivalrous, because if I eat an apple, you cannot eat the same apple. 

A printed newspaper may be rivalrous, because it may not be easy for you and me to hold the same sheet of paper and read it at the same time. However, the ''news'' itself is non-rivalrous, because both of us and anyone else can consume the same news at the same time, once it is produced, especially if it's published openly on the Internet or broadcasted on radio or television. 

Non-excludable means that once the good is produced, anyone can use it without paying for it. Information is non-excludable, because everyone can consume it at the same time once it becomes available. [[w:Copyright|Copyright]] law does ''not'' apply to information: It applies to ''expression''.&lt;ref&gt;The US Copyright Act of 1976, Section 102, says, &quot;Copyright protection subsists ... in original works of authorship fixed in any tangible medium of expression ... . In no case does copyright protection ... extend to any idea, procedure, process, system, method of operation, concept, principle, or discovery.&quot; 17 U.S. Code § 102. &lt;!--US Copyright Law of 1976--&gt;{{cite Q|Q3196755}}&lt;/ref&gt; [[w:Joseph Stiglitz|Stiglitz]] (1999) said that [[w:Thomas Jefferson|Thomas Jefferson]] anticipated the modern concept of information as a public good by saying, &quot;He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me.&quot; Stiglitz distinguished between &quot;push and pull mechanisms&quot; to promote innovation and creative work: &quot;Push&quot; mechanisms pay for work upfront, hoping that it will achieve a desired outcome, like citizen-directed subsidies for newspapers. &quot;Pull&quot; mechanisms set a target and then reward those who reach the target, like copyrights and patents.&lt;ref&gt;Baker (2023).&lt;/ref&gt;

Lindahl (1919, 1958) recommended taxing people for public goods in proportion to the benefits they receive. For subsidies for news, especially citizen-directed, this would mean taxing primarily the poor and middle class to fund this.&lt;ref&gt;For more on this, see the Wikipedia articles on [[w:Lindahl tax|Lindahl tax]] and [[w:Theories of taxation|Theories of taxation]].&lt;/ref&gt; If they receive benefits as claimed in the literature cited in this article, the benefits they receive would soon exceed the taxes they pay for it, making the news subsidies effectively free in perpetuity, paid by benefits the poor and middle class would not have without these subsidies. If Piketty (2021, cited below with Figure 1) is correct, the ultra-wealthy would likely also benefit in absolute terms, though the relative distinction between them and the poor would be reduced.

This article recommends [[w:randomized controlled trials|randomized controlled trials]] to quantify the extent to which experimental interventions benefit the public by modifying information environment(s) in ways that (a) reduce political polarization and / or violence and / or (b) improve broadly shared peace and prosperity for the long term.

=== Sharing increases the value === 

The logic behind claiming that &quot;information is a public good&quot; can be easily understood as follows:  

:''If I know the best solution to any major societal problem, it will not help anyone unless a critical mass of some body politic shares that perception.  Conversely, if a critical mass of a body politic believes in the need to implement a certain reform, it will happen, even if I am ignorant of it or completely opposed to it.''  

We can extend this analysis to our worst enemies:  

:It is in ''our best interest'' to help people supporting our worst enemies get information they want, ''independent'' of controls that people with power exercise over nearly all major media today:  If our actions reduce the ability of their leaders to censor their media (and of our political and economic leaders to censor ours), the information everyone gets should make it harder for leaders to convince others to support measures contrary to nearly everyone's best interests.  

What kinds of data can we collect and analyze to evaluate who benefits and who loses from alternative interventions attempting to improve the media? See below.&lt;ref&gt;The power relationship between media and politicians can go both ways.  In addition to asking the extent to which politicians control the media, we can also consider the extent to which political leaders might feel constrained by the major media: To what extent do the major media create the stage upon which politicians read their lines, as claimed in the Wikiversity article on &quot;[[Confirmation bias and conflict]]&quot;?  Might a more diverse media environment make it easier for political leaders to pursue policies informed more by available research and less by propaganda?  Might experiments as described herein help politicians develop more effective governmental policies, because of a reduction in the power of media whose ownership and funding are more diverse? This is discussed further in this article in a section on [[Information is a public good: Designing experiments to improve government#Media and war|Media and war]].&lt;/ref&gt;

=== World Bank on the value of information === 

In 2002 the President of the [[w:World Bank|World Bank]],&lt;ref&gt;The 2022 World Bank Group portfolio was 104 billion USD (World Bank 2022, Table 1, p. 13; 17/116 in PDF).  An improvement of 0.1 percentage points in the performance of that portfolio would be 104 million.  A lot could be accomplished with budgets much smaller than this.&lt;/ref&gt; [[w:James Wolfensohn|James Wolfensohn]], wrote, &quot;[A] free press is not a luxury. It is at the core of equitable development. The media can expose corruption. ... They can facilitate trade [and bring] health and education information to remote villages ... . But ... the independence of the media can be fragile and easily compromised. All too often governments shackle the media. Sometimes control by powerful private interests restricts reporting. ... [T]o support development, media need the right environment{{mdash}}in terms of freedoms, capacities, and checks and balances.&quot;&lt;ref&gt;Wolfensohn (2002). More on this is available in other contributions to Islam et al. (2002) including [[w:Joseph Stiglitz|Stiglitz]] (2002), who noted the following: &quot;There is a natural asymmetry of information between those who govern and those whom they are supposed to serve. ... Free speech and a free press not only make abuses of governmental powers less likely, they also enhance the likelihood that people's basic social needs will be met. ... [S]ecrecy distorts the arena of politics. ... Neither theory nor evidence provides much support for the hypothesis that fuller and timelier disclosure and discussion would have adverse effects. ... The most important check against abuses is a competitive press that reflects a variety of interests. ... [F]or government officials to appropriate the information that they have access to for private gain ... is as much theft as stealing any other public property.&quot;&lt;/ref&gt;

Below please find proposals for evaluating alternative ways of improving the media and circumstances under which they are or are not effective.

=== US Postal Service Act of 1792: a natural experiment === 

[[w:Robert W. McChesney|McChesney]] and [[w:John Nichols (journalist)|Nichols]] (2010, 2016) suggested that the US [[w:Postal Service Act|Postal Service Act]]&lt;ref&gt;Wikipedia &quot;[[w:Postal Service Act|Postal Service Act]]&quot;, accessed 2023-07-11.&lt;/ref&gt; of 1792 made a major contribution to making the US what it is today.  Under that act, newspapers were delivered up to 100 miles for a penny, when first class postage was between 6 and 25 cents depending on distance.  They estimated that between 1840 and 1844, the US postal subsidy was 0.211 percent of GDP with federal printing subsidies adding another 0.005 percent, totaling 0.216 percent of GDP,&lt;ref name=McC-N2010&gt;McChesney and Nichols (2010, pp. 310-311, note 88).&lt;/ref&gt; roughly $140 per person per year in 2019 dollars.&lt;ref name=McN_IMF&gt;International Monetary Fund (2023):  US Gross domestic product per capita at current prices was estimated at $65,077 for 2019 on 2023-04-28. 0.211% of $65,077 = $137; round to $140 for convenience.&lt;/ref&gt; We use 2019 dollars here to make it easy to compare with Rolnik et al. (2019), who recommended $50 per adult per year, which is roughly 0.08 percent of US GDP.  Rolnik et al. added that the level of subsidies would require &quot;extensive deliberation and experimentation&quot;.&lt;ref name=Rolnik&gt;Rolnik et al. (2019, p. 44).  Per [[w:Demographics of the United States]], 24 percent of the US population is under 18, so adults are 76 percent of the population. Thus, $50 per adult is $37.50 per capita. US GDP per capita in $65,0077 in 2019 in current dollars per International Monetary Fund (2023). Thus, $37.50 per capita would be roughly 0.077 percent of GDP; round to 0.08 percent for convenience.&lt;/ref&gt; More recently McChesney and Nichols have recommended 0.15 percent of GDP, considering the fact that the advent of the Internet has nearly eliminated the costs of printing and distribution.&lt;ref name=McC-N2021&gt;McChesney and Nichols (2021; 2022, p. 19).&lt;/ref&gt;

[[w:Alexis de Tocqueville|Tocqueville]], who visited the US in 1831, observed the following:  
* [T]he liberty of the press does not affect political opinion alone, but extends to all the opinions of men, and modifies customs as well as laws. ... I approve of it from a consideration more of the evils it prevents, than of the advantages it insures.&lt;ref&gt;Tocqueville (1835; 2001, p. 91).  In 2002 Roumeen Islam stated this more forcefully:  &quot;Arbitrary actions by government are always to be feared. If there is to be a bias in the quantity of information that is released, then erring on the side of more freedom rather than less would appear to cause less harm.&quot; (World Bank, 2002, pp. 21-22; 33-34/336 in pdf).&lt;/ref&gt;
* The liberty of writing ... is most formidable when it is a novelty; for a people who have never been accustomed to hear state affairs discussed before them, place implicit confidence in the first tribune who presents himself.  The Anglo-Americans have enjoyed this liberty ever since the foundation of the Colonies ... .  A glance at a French and an American newspaper is sufficient to show the difference ... .  In France, the space allotted to commercial advertisements is very limited, and the news-intelligence is not considerable; but the essential part of the journal is the discussion of the politics of the day.  In America, three-quarters of the enormous sheet are filled with advertisements, and the remainder is frequently occupied by political intelligence or trivial anecdotes:  it is only from time to time that one finds a corner devoted to passionate discussions, like those which the journalists of France every day give to their readers.&lt;ref&gt;Tocqueville (1835; 2001, p. 92).&lt;/ref&gt; 
* It has been demonstrated by observation, and discovered by the sure instinct even of the pettiest despots, that the influence of a power is increased in proportion as its direction is centralized.&lt;ref&gt;Tocqueville (1835; 2001, pp. 92-93).&lt;/ref&gt;
* [T]he number of periodical and semi-periodical publications in the United States is almost incredibly large. In America there is scarcely a hamlet which does not have its newspaper.&lt;ref&gt;Tocqueville (1835; 2001, p. 93).&lt;/ref&gt;
* In the United States, each separate journal exercises but little authority; but the power of the periodical press is second only to that of the people ... .&lt;ref&gt;Tocqueville (1835; 2001, p. 94). &lt;/ref&gt;

[[File:Real US GDP per capita in 5 epocs.svg|thumb|Figure 1.  Average annual income (Gross Domestic Product per capita adjusted for inflation) in the US 1790-2021 showing five epochs identified in a &quot;breakpoint&quot; analysis (to 1929, 1933, 1945, 1947, 2021) documented in the Wikiversity article on &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;.&lt;ref&gt;Wikiversity &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;, accessed 2023-07-18.&lt;/ref&gt; Piketty (2021, p. 139) noted, &quot;In the United States, the national income per inhabitant rose at a rate ... of 2.2 percent between 1950 to 1990 when the top tax rate reached on average 72 percent.  The top rate was then cut in half, with the announced objective of boosting growth.  But in fact, growth fell by half, reaching 1.1 percent per annum between 1990 and 2020&quot;.&lt;ref&gt;A more recent review of the literature of the impact of inequality on growth is provided by Jahangir (2023, sec. 3), who notes that some studies have claimed that inequality ''increases'' the rate of economic growth, while other reach the opposite conclusion. However, 'the preponderant academic position is shifting from the argument that “we don’t have enough evidence” and towards seriously addressing and combating economic inequality.'&lt;/ref&gt; Our analysis of US GDP per capita from Measuring Worth do not match Piketty's report exactly, but they are close. We got 2.3 percent annual growth from 1947 to 1990 then 1.8 percent to 2008 and 1.1 percent to 2020. However, we have so far been unable to find a model that suggests that this decline is statistically significant.]]  

To what extent was [[w:Alexis de Tocqueville|Tocqueville's]] &quot;incredibly large&quot; &quot;number of periodical and semi-periodical publications in the United States&quot; due to the US Postal Service Act of 1792? To what extent did that &quot;incredibly large&quot; number of publications encourage literacy, limit political corruption, and help the US of that day remain together and grow both in land area and economically while contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically? To what extent does the enormous power of the US today rest on the economic growth of that period and its impact on the political culture of that day continuing to the present?&lt;ref&gt;Wikiversity &quot;[[The Great American Paradox]]&quot;, accessed 2023-06-12.&lt;/ref&gt; That growth transformed the US into the world leader that it is today;  see Figure 1.  In the process, it generated new technologies that benefit the vast majority of the world's population alive today. If the newspapers Tocqueville read made any substantive contribution to the growth summarized in Figure 1, the information in those newspapers were public goods potentially benefiting the vast majority of humanity ''to the end of human civilization.''&lt;ref&gt;Acemoglu (2023) documents how the power of monopolies and other politically favored groups often distorts the direction of technology development into suboptimal technologies.  Might increasing the funding for more independent news outlets reduce the power of such favored groups and thereby help correct these distortions and deliver &quot;sizable welfare benefits&quot;, e.g., &quot;in the context of industrial automation, health care, and energy&quot;?&lt;/ref&gt; Experiments of the type discussed below can help quantify the magnitude of these suggested benefits in contemporary settings.

=== Other economists === 

We cannot prove that the diversity of newspapers in the early US contributed to the economic growth it experienced. Banerjee and Duflo (2019) concluded that no one knows how to create economic growth. They won the 2019 Nobel Memorial Prize in Economics with Michael Kremer for their leadership in using [[w:randomized controlled trials|randomized controlled trials]]&lt;ref&gt;Wikipedia &quot;[[w:Randomized controlled trials|Randomized controlled trials]]&quot;, accessed 2023-07-11.&lt;/ref&gt; to learn how to reduce global poverty.&lt;ref&gt;Wikipedia &quot;[[w:2019 Nobel Memorial Prize in Economic Sciences|2019 Nobel Memorial Prize in Economic Sciences]]&quot;, accessed 2023-06-13.  Nobel Prize (2019).  Amazon.com indicates that distribution of the book started 2019-11-12, twenty-nine days after the Nobel prize announcement 2019-10-14. Evidently the book must have been completed before the announcement.&lt;/ref&gt; More recently, Wake et al. (2021) found evidence that ''the economic costs of curbing press freedom persist long after such freedoms have been restored.''&lt;ref&gt;See also Nguyen et al. (2021).&lt;/ref&gt; And Mohammadi et al. (2022) found that economic growth rates were impacted by civil liberties, economic and press freedom and the economic growth rates of neighbors (spacial autocorrelation) but not democracy. These findings of Mohammidi et al. (2022) and Wake et al. (2021) reinforce Thomas Jefferson's 1787 comment that, &quot;were it left to me to decide whether we should have a government without newspapers, or newspapers without a government, I should not hesitate a moment to prefer the latter.&quot;&lt;ref&gt;From a letter to Colonel Edward Carrington (16 January 1787), cited in Wikiquote, &quot;[[Wikiquote:Thomas Jefferson|Thomas Jefferson]]&quot;, accessed 2023-07-29.&lt;/ref&gt; To what extent might experiments like those recommended in this article either reinforce or refute this claim of Jefferson from 1787?

=== Randomized controlled trials to quantify the value of information === 

This article suggests randomized controlled trials to quantify the impact of citizen-directed subsidies for journalism, roughly following the recommendations of McChesney and Nichols (2021, 2022) to distribute some small percentage of GDP to local news nonprofits ''via local elections''.  Philanthropies could fund such experiments for some of the smallest and poorest places in the world.  Organizations like the World Bank could fund such experiments as adjuncts to a random selection from some list of other interventions they fund, justified for the same reason that they would not consider funding anything without appropriate accounting and auditing of expenditures, as discussed further below.  

Before making suggestions regarding experiments, we review previous research documenting how information might be a public good.

== Previous research ==

Before considering optimal level of subsidies for news, it may be useful to consider the research for which [[w: Daniel Kahneman|Daniel Kahneman won the 2002 Nobel Memorial Prize in Economics]].&lt;ref&gt;Wikipedia &quot;[[w:Daniel Kahneman|Daniel Kahneman]]&quot;, accessed 2023-04-28.&lt;/ref&gt; Most important for present purposes may be that virtually everyone: 

# thinks they know more than they do ([[w:Overconfidence effect|Overconfidence]]),&lt;ref&gt;Wikipedia &quot;[[w:Overconfidence effect|Overconfidence effect]]&quot;, accessed 2023-04-29. Kahneman and co-workers have documented that experts are also subject to overconfidence and in some cases may be worse.  Kahneman and Klein (2009) found that expert intuition can be learned from frequent, rapid, high-quality feedback about the quality of their judgments.  Unfortunately, few fields have that much quality feedback.  Kaheman et al. (2021) call practitioners with credentials but without such expert intuition &quot;respect-experts&quot;.  Kahneman (2011, p. 234) said his &quot;most satisfying and productive adversarial collaboration was with Gary Klein&quot;.&lt;/ref&gt; and 
# prefers information and sources consistent with preconceptions. ([[w:Confirmation bias|Confirmation bias]]).&lt;ref&gt;Wikipedia &quot;[[w:Confirmation bias|Confirmation bias]]&quot;, accessed 2023-04-29.&lt;/ref&gt;

To what extent do media organizations everywhere exploit the confirmation bias and overconfidence of their audience to please those who control the money for the media, and to what extent might this ''reduce'' broadly shared economic growth?  The proposed experiments should include efforts to quantify this, measuring, e.g., local incomes, inequality, political polarization and the impact of interventions attempting to improve such.  

Plous wrote, &quot;No problem in judgment and decision making is more prevalent and more potentially catastrophic than overconfidence.&quot;&lt;ref&gt;Plous (1993, p. 217).  See also Wikipedia &quot;[[w:Overconfidence effect|Overconfidence effect]]&quot;, accessed 2023-04-29.&lt;/ref&gt; It contributes to inordinate losses by all parties in negotiations of all kinds&lt;ref&gt;Thompson (2020).&lt;/ref&gt; including lawsuits,&lt;ref&gt;Loftus and Wagenaar (1988).&lt;/ref&gt; strikes,&lt;ref&gt;Babcock and Olson (1992) and Thompson and Loevenstein (1992).&lt;/ref&gt; financial market bubbles and crashes,&lt;ref&gt;Daniel et al. (1998).&lt;/ref&gt; and politics and international relations,&lt;ref&gt;&lt;!-- Dominic D.P. Johnson (2020) Strategic instincts: the adaptive advantages of cognitive biases in international politics--&gt;{{cite Q|Q120967807}}&lt;/ref&gt; including wars.&lt;ref&gt;Johnson (2004).&lt;/ref&gt;

Might the frequency and expense of lawsuits, strikes, financial market volatility, political coruption and wars be reduced by encouraging people to get more curious and search more often for information that might contradict their preconceptions?  Might such discussions be encouraged by interventions such as increasing the total funding for news through many small, independent, local news organizations?  If yes, to what extent might such experimental interventions threaten the hegemony of major media everywhere while benefitting everyone, with the possible exception of those who benefit from current systems of political corruption?  

[[File:Knowledge v. public media.png|thumb|Figure 2.  Knowledge v. public media: Percent correct answers in surveys of knowledge of domestic and international politics vs. per capita subsidies for public media in Denmark (DK), Finland (FI), the United Kingdom (UK) and the United States (US).&lt;ref&gt;&quot;politicalKnowledge&quot; dataset in Croissant and Graves (2022), originally from ch. 1, chart 8, p. 268 and ch. 4, chart 1, p. 274, McChesney and Nichols (2010).&lt;/ref&gt;]]

One attempt to quantify this appears in Figure 2, which summarizes a natural experiment on the impact of government subsidies for public media on public knowledge of domestic and international politics: Around 2008 the governments of the US, UK, Denmark and Finland provided subsidies of $1.35, $80, $101 and $101 per person per year, respectively, for public media.  A survey of public knowledge of domestic and international politics found that people with college degrees seemed to be comparably well informed in the different countries, but people with less education were better informed in the countries with higher public subsidies.  

Kaviani et al. (2022) studied the impact of &quot;the staggered expansion of [[w:Sinclair Broadcast Group|Sinclair Broadcast Group]]: the largest conservative network in the U.S.&quot; They documented a decline in Corporate Social Responsibility (CSR) ratings of firms headquartered in Sinclair expansion areas.  They also documented a &quot;right-ward ideological shift&quot; in coverage that was &quot;nearly one standard deviation of the ideology distribution&quot; as well as &quot;substantial decreases in coverage of local politics substituted by increases in national politics.&quot; Ellison (2024) said that &quot;Sinclair's recipe for TV news&quot; includes an annual survey asking viewers, &quot;What are you most afraid of?&quot; Sinclair reportedly focuses on that while implying in their coverage &quot;that America's cities, especially those run by Democratic politicians, are dangerous and dysfunctional.&quot; 

Sources in France are concerned that billionaire [[w:Vincent Bolloré|Vincent Bolloré]] has purchased a substantial portion of French media and used it effectively to promote the French far right.&lt;ref&gt;Francois (2022). Cagé (2022). Cagé and Stetler (2022).&lt;/ref&gt; Scheidler (2024a) reported that the concentration of ownership the German media &quot;has not yet reached the extreme forms observed in France, the United Kingdom or the United States, but the process of consolidation initiated several decades ago has transformed a landscape renowned for its decentralization.&quot;&lt;ref&gt;Translated from, &quot;la concentration de la propriété dans la presse suprarégionale n’a pas encore atteint les formes extrêmes observées en France, au Royaume-Uni ou aux États-Unis, mais le processus de consolidation enclenché depuis plusieurs décennies a transformé un paysage réputé pour sa décentralisation.&quot;&lt;/ref&gt;&lt;ref&gt;See also ''Die Tageszeitung'' (2023).&lt;/ref&gt; Scheidler (2024b) reported that there still exists a wide range of constructive media criticism in Germany, but it gets less coverage than before in the increasingly consolidated major media. This has driven many who are not happy with these changes to alternative media such as ''[[w:Die Tageszeitung|Die Tageszeitung]]'', founded in 1978.

Benton wrote that past research has shown that strong local newspapers &quot;increase voter turnout, reduce government corruption, make cities financially healthier, make citizens more knowledgeable about politics and more likely to engage with local government, force local TV to raise its game, encourage split-ticket (and thus less uniformly partisan) voting, make elected officials more responsive and efficient ... And ... you get to reap the benefits of all those positive outcomes ''even if you don’t read them yourself''.&quot;&lt;ref&gt;Benton (2019); italics in the original.  See also Green et al. (2023, p. 7), Schulhofer-Wohl and Garrido (2009) and Stearns and Schmidt (2022).  A not quite silly example of this is documented in the Wikipedia article on the &quot;[[w:City of Bell scandal|City of Bell scandal]]&quot; accessed 2023-05-05:  Around 1999 the local newspaper died.  In 2010 the ''[[w:Los Angeles Times|Los Angeles Times]]'' reported that the city was close to bankruptcy in spite of having atypically high property tax rates.  The compensation for the City Manager was almost four times that of the President of the US, even though Bell, California, had a population of only approximately 38,000.  The Chief of Police and most members of the City Council also had exceptionally high compensations.  It was as if the City Manager had said in 1999, &quot;Wow:  The watchdog is dead.  Let's have a party.&quot;&lt;/ref&gt;

We feel a need to repeat that last comment:  ''You and I'' benefit from others consuming news that we do not, because they become less likely to be stampeded into voting contrary to their best interests{{mdash}}and ours{{mdash}}and more likely to lobby effectively against questionable favors to major political campaign contributors or other people with power, underreported by major media that have conflicts of interest in balanced coverage of anything that might offend people with substantive control of their funding.  That suggests that everyone might benefit from subsidizing ''a broad variety of independent'' local news outlets consumed by others.&lt;ref&gt;Some of those who benefit from the current system of political corruption may lose from the increased transparency produced by increases in the quality, quantity, diversity, and broader consumption of news.  However, Bezruchka (2023) documents how even the ultra-wealthy in countries with high inequality generally have shorter life expectancies than their counterparts in more egalitarian societies:  What they might lose in social status would likely be balanced by a reduction in stress and exposure to life-threatening incidents.&lt;/ref&gt; Experiments along the lines discussed below could attempt to evaluate these claims and estimate their magnitudes.

== How fair is the US tax system? ==

How fair is the US tax system?  It depends on who is asked and how fairness is defined.  

[[File:Share of taxes vs. AGI.svg|thumb|Figure 3.  Effective tax rate vs. Adjusted Gross Income (AGI).&lt;ref&gt;York (2023) based on analyses published by the US Internal Revenue Service (IRS).&lt;/ref&gt;]]

The [[w:Tax Foundation|Tax Foundation]] computed the effective tax rate in different portions of the distribution of Adjusted Gross Income (AGI), plotted in Figure 3.  They noted that,&quot;half of taxpayers paid 99.7 percent of federal income taxes&quot;. The effective tax rate on the 1 percent highest adjusted gross incomes (AGIs) was 26 percent, almost double (1.91 times) the average, while the effective tax rate for the bottom half was 3.1 percent, only 23 percent of the average.&lt;ref&gt;York (2023).&lt;/ref&gt;

The Tax Foundation did ''not'' mention that we get a very different perspective from considering ''gross income'' rather than AGI. Leiserson and Yagan (2021)&lt;ref&gt;published by the Biden White House.&lt;/ref&gt; estimated that the average ''effective'' federal individual income tax rate paid by America’s 400 wealthiest families&lt;ref&gt;The &quot;400 wealthiest families&quot; are identified in &quot;[[w:The Forbes 400|The Forbes 400]]&quot;; see the Wikipedia article with that title, accessed 2023-05-07.&lt;/ref&gt; was between 6 and 12 percent with the most likely number being 8.2 percent. The difference comes in the ''adjustments'', while the uncertainty comes primarily from appreciation in the value of unsold stock,&lt;ref&gt;Unsold stock or other property subject to capital gains tax, which in 2022 was capped at 20 percent;  see Wikipedia, &quot;[[w:Capital gains tax in the United States|Capital gains tax in the United States]]&quot;, accessed 2023-05-08.&lt;/ref&gt; which is taxed at a maximum of 20 percent when sold and never taxed if passed as inheritance.&lt;ref&gt;The Wikipedia article on &quot;[[w:Estate tax in the United States|Estate tax in the United States]]&quot; describes an &quot;Exclusion amount&quot;, which is not taxed in inheritance.  That exclusion amount was $675,000 in 2001 and has generally trended upwards since except for 2010, and was $12.06 million in 2022. (accessed 2023-05-08.)&lt;/ref&gt;

Divergent claims about ''business'' taxes can similarly be found.  Watson (2022) claimed  that, &quot;Corporate taxes are one of the most economically damaging ways to raise revenue and are a promising area of reform for states to increase competitiveness and promote economic growth, benefiting both companies and workers.&quot;  His &quot;economically damaging&quot; claim seems contradicted by the claim of Piketty (2021, p. 139), cited with Figure 1 above, that when the top income tax rate was cut in half, the rate of economic growth in the US fell by half, instead of increasing as Watson (2022) suggested. By contrast, Fuhrmann and Uradu (2023) describe, &quot;How large corporations avoid paying taxes&quot;.   

[[File:UStaxWords.svg|thumb|Figure 4.  Millions of words in the US federal tax code and regulations, 1955-2015, according to the [[w:Tax Foundation|Tax Foundation]]. [1=income tax code; 2=other tax code; 3=income tax regulations; 4=other tax regulations; solid line= total]&lt;ref&gt;&quot;UStaxWords&quot; dataset in Croissant and Graves (2022) from the Tax Foundation.&lt;/ref&gt;]]

One reference on the difference between &quot;adjusted&quot; and &quot;gross&quot; income is US federal tax code and regulations, which grew from 1.4 million words in 1955 to over 10 million in 2015, averaging 145,000 additional words each year;  see Figure 4.  

How does this relate to media?

== How do media organizations make money? ==

Media organizations everywhere sell changes in audience behaviors to the people who give them money.  If they do not have an audience, they have nothing to sell.  If they sufficiently offend their funders, they will not get the revenue needed to produce content.&lt;ref&gt;A famous illustration of this conflict between content and funding was when CBS Chairman [[w:William S. Paley|William Paley]] reportedly told [[w:Edward R. Murrow|Edward R. Murrow]] in 1958 that he was discontinuing Murrow's award-winning show ''[[w:See It Now|See It Now]]'', because &quot;I don't want this constant stomach ache every time you do a controversial subject&quot;, documented in Friendly (1967, p. 92).&lt;/ref&gt;  

The major media in the US have conflicts of interest in honestly reporting on discussions in congress on copyright law or on anything that might impact a major advertiser or might make it easier for politicians to get elected by spending less money on advertising.  McChesney (2015) insisted that the major media are not interested in providing information that people want:  They are interested in making money and protecting the interests of the ultra-wealthy, who control the largest advertising budgets.  For example, media coverage of the roughly 40,000 people who came to [[w:1999 Seattle WTO protests|Seattle in 1999 to protest the WTO]] Ministerial Conference there&lt;ref&gt;Wikipedia &quot;[[w:1999 Seattle WTO protests|1999 Seattle WTO protests]]&quot;, accessed 2023-05-08.&lt;/ref&gt; and the 10,000 - 15,000 who came to [[w:Washington A16, 2000|Washington, DC, the following year]] to protest the International Monetary Fund and the World Bank,&lt;ref&gt;Wikipedia &quot;[[w:Washington A16, 2000|Washington A16, 2000]]&quot;, accessed 2023-05-08.&lt;/ref&gt; included &quot;some outstanding pieces produced by the corporate media, but those were exceptions to the rule. ... [T]he closer a story gets to corporate power and corporate domination of our society, the less reliable the corporate news media is.&quot;&lt;ref&gt;McChesney (2015, p. xx).&lt;/ref&gt; 

Aaron (2021) said, &quot;Bob McChesney ... taught me [to] look at ... the stories that are cheap to cover.&quot; Between around 1975 and 2000, the major commercial broadcasters in the US fired nearly all their investigative journalists&lt;ref&gt;McChesney (2004, p. 81):  &quot;A five-year study of investigative journalism on TV news completed in 2002 determined that investigative journalism has all but disappeared from the nation's commercial airwaves.&quot;&lt;/ref&gt; and replaced them with the police blotter.  It's easy and cheap to repeat what the police say.&lt;ref&gt;Holmes (2022) quoted Ryan Sorrell, Founder and Publisher of the Kansas City Defender, as saying, &quot;the media often parrots or repeats what police and news releases say.&quot;&lt;/ref&gt; A news outlet can do that without seriously risking loss of revenue.  In addition, poor defendants who may not have money for legal defense will rarely have money to sue a media outlet for defamation.  By contrast, a news report on questionable activities by a major funder risks both direct loss of advertising revenue and being sued.&lt;ref&gt;The risks of being sued include the risks of [[w:Strategic lawsuit against public participation|strategic lawsuits against public participation]] (SLAPPs) by major organizations, which can intimidate journalists and publishers as well as potential whistleblowers, who might inform journalists of violations of law by their employers.  Some of these are documented in the &quot;[[w:Freedom of the press in the United States#U.S. Press Freedom Tracker|U.S. Press Freedom Tracker]]&quot;, maintained by the [[w:Freedom of the Press Foundation|Freedom of the Press Foundation]] and the [[w:Committee to Protect Journalists|Committee to Protect Journalists]].  These include arrests, assaults, threats, denial of access, equipment damage, prior restraint, and subpoenas which could intimidate journalists, publishers, and employees feeling a need to expose violations of law and threats to public safety.  See Wikipedia &quot;[[w:Freedom of the Press Foundation|Freedom of the Press Foundation]]&quot;, &quot;[[w:Committee to Protect Journalists|Committee to Protect Journalists]]&quot;, and &quot;[[w:Strategic lawsuit against public participation|Strategic lawsuit against public participation]]&quot;, accessed 2023-07-11.&lt;/ref&gt;  These risks impose a higher standard of journalism (and additional costs) when reporting on questionable activities by people with power than when reporting on poor people.  This is a much bigger problem in countries where libel is a criminal rather than a civil offense or where truth is not a defense for libel.&lt;ref&gt;Islam et al. (2002), esp. pp. 12-13 (24-25/336 in pdf), p. 50 (62/336 in pdf), and ch. 11, pp. 207-224 (219-236/336 in pdf). [[w:United States defamation law|Truth was not a defense against libel in the US]] in 1804 when Harry Croswell lost in ''[[w:United States defamation law#People v. Croswell|People v. Croswell]]''.  That began to change the next year when the [[w:United States defamation law#People v. Croswell|New York State Legislature]] changed the law to allow truth as a defense against a libel charge. Seventy years earlier in 1735 [[w:John Peter Zenger#Libel case|John Peter Zenger]] was acquitted of a libel charge, but only by [[w:Jury nullification in the United States|jury nullification]].&lt;/ref&gt;

[[File:U.S. incarceration rate since 1925.svg|thumb|Figure 5.  Percent of the US population in state and federal prisons  [male (dashed red), combined (solid black), female (dotted green)]&lt;ref&gt;&quot;USincarcerations&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

After about 1975 the public noticed the increased coverage of crime in the broadcast news and concluded that crime was out of control, when there had been no substantive change in crime.  They voted in a generation of politicians, who promised to get tough on crime.  The incarceration rate in the US went from 0.1 percent to 0.5 percent in the span of roughly 25 years, after having been fairly stable for the previous 50 years;  see Figure 5.&lt;ref&gt;Potter and Kapeller (1998). Sacco (1998, 2005).&lt;/ref&gt;  

[[File:IncomeInequality9b.svg|thumb|Figure 6.  Average and quantiles of family income (Gross Domestic Product per family) in constant 2010 dollars.&lt;ref&gt;&quot;incomeInequality&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

Around that same time, income inequality in the US began to rise;  see Figure 6.&lt;ref&gt;Bezruchka (2023) summarizes research documenting how &quot;inequality kills us all&quot;. He noted that the US was among the leaders in infant mortality and life expectancy in the 1950s.  Now the US is trailing most of the advanced industrial democracies per United Nations (2022).  He attributes the slow rate of improvement in public health in the US to increases in inequality.  That argument is less than perfect, because Figure 5 suggests that inequality in the US did not begin to increase until around 1975, but the divergence in public health between the US and other advanced industrial democracies seems more continuous between the 1950s and the present, 2023.  Beyond this, Graves and Samuelson (2022) noted that it is in everyone's best interest to help others with conditions that might be infectious to get competent medical assistance, because that reduces our risk of contracting their disease and possibly dying from it. Bezruchka (2023) cites documentation claiming that even the wealthy in the US have lower life expectancy than their counterparts in other advanced industrial democracies, because the high level of inequality in the US means that the ultra-wealthy in the US get exposed to more pathogens than their counterparts elsewhere.  See also Wilkinson and PIckett (2017).&lt;/ref&gt; To what extent might that increase in inequality be due to the structure of the major media?  

To what extent might you and I benefit from making it easier for millions of others to research different aspects of government policies including the &quot;adjustments&quot; in the US tax system embedded in the over 10 million words of US federal tax code and regulations documented with Figure 4 above, encouraging them to lobby the US Congress against the special favors granted to major political campaign contributors against the general welfare of everyone else? Everyone except the beneficiaries of such political corruption would likely benefit from the news that helps concerned citizens lobby effectively against such corruption, even if we did not participate in such citizen lobbying efforts and were completely ignorant of them.

== Media and war == 

:[[w:You've Got to Be Carefully Taught|''&quot;You've got to be taught to hate and fear. ... It's got to be drummed in your dear little ear.&quot;'']] 
:Lt. Cable in the [[w:South Pacific (musical)|1949 Rodgers and Hammerstein musical ''South Pacific'']].  

To what extent is it accurate to say that before anyone is killed in armed hostilities, the different parties to the conflict are polarized by the different media the different parties find credible?&lt;ref&gt;The role of the media in war has long been recognized.  It is commonly said that the first casualty of war is truth. Knightley (2004, p. vii) credits Senator Hiram Johnson as saying in 1917, &quot;The first casualty when war comes is truth.&quot; However, the Wikiquote article on &quot;[[Wikiquote:Hiram Johnson|Hiram Johnson]]&quot; says this quote has been, &quot;Widely attributed to Johnson, but without any confirmed citations of original source. ... [T]he first recorded use seems to be by Philip Snowden.&quot; (accessed 2023-07-22.)&lt;/ref&gt; 

This might seem obvious, but how can we quantify political polarization in a way (a) that correlates with the severity of the conflict and (b) can be used to evaluate the effectiveness of efforts to reduce the polarization? 

The [[w:International Panel on the Information Environment|International Panel on the Information Environment]] (IPIE) is a consortium of over 250 global experts developing tools to combat political polarization driven by the structure of the Internet.&lt;ref&gt;e.g., National Acadamies (2023).&lt;/ref&gt; The US Institute of Peace (2016) discusses &quot;Tools for Improving Media Interventions in Conflict Zones&quot;. Previous research in this area was summarized by Arsenault et al. (2011). One such tool might be video games.&lt;ref&gt;Caelin (2016).&lt;/ref&gt;

We suggest experimenting with interventions designed to reduce political polarization with some of the smallest but most intense conflicts: Interventions that require money could be more effectively tested with smaller, high intensity conflicts. With randomized controlled trials, it would be easier to measure a reduction from a higher-intensity conflict, and a smaller population could commit more money per capita with a relatively modest budget.

The [[w:Armed Conflict Location and Event Data Project|Armed Conflict Location and Event Data Project (ACLED)]] tracks politically relevant violent and nonviolent events by a range of state and non-state actors. Their data can help identify countries or geographic regions in conflict as candidates to be [[w:Randomized controlled trial|assigned randomly to experimental and control groups]], whose comparison can provide high quality data to help evaluate the impact of any intervention. Initial experiments of this nature might be done with a modest budget by working with organizations advocating nonviolence and with religious groups to recruit diaspora communities to do things recommended by experts in IPIE while also lobbying governments for funding. Any success can be leveraged into changes in foreign and military policies to make the world safer for all. 

Before discussing such experiments further, we consider a few case studies. 

=== Russo-Ukrainian War and the US Civil War === 

In the [[w:Russo-Ukrainian War|Russo-Ukrainian War]], Halimi and Rimbert (2023) describe &quot;Western media as cheerleaders for war&quot;. [[w:Joseph Stiglitz|Stiglitz]] (2002) noted this was a general phenomenon:  &quot;In periods of perceived conflict ... a combination of self-censorship and reader censorship may also undermine the ability of a supposedly free press to ensure democratic transparency and openness.&quot; Media organizations do not always do this solely to please their funders.  Reporters are killed&lt;ref&gt;Different lists of journalists killed for their work are maintained by the [[w:Committee to Protect Journalists|Committee to Protect Journalists]], (CPJ), [[w:Reporters Without Borders|Reporters without Borders]], and the [[w:International Federation of Journalists|International Federation of Journalists]].  CPJ has claimed that their numbers are typically lower, because their confirmation process may be more rigorous.  See Committee to Protect Journalists (undated) and the Wikipedia articles on &quot;[[w:Committee to Protect Journalists|Committee to Protect Journalists]]&quot;, &quot;[[w:Reporters Without Borders|Reporters without Borders]]&quot;, and the &quot;[[w:International Federation of Journalists|International Federation of Journalists]]&quot;, accessed 2023-07-11.&lt;/ref&gt; or jailed and news outlets closed to prevent them from disseminating information that people with power do not want distributed.  Early in the Civil War in the US (1861-1865), some newspapers in the North said the US should let the South secede, because that would be preferable to war.  Angry mobs destroyed some offices and printing presses.  One editor &quot;was forcibly taken from his house by an excited mob, ... covered with a coat of tar and feathers, and ridden on a rail through the town.&quot; Others changed their policies &quot;voluntarily&quot;, recognizing threats to their lives or property or to a loss of audience.&lt;ref&gt;Harris (1999, esp. p. 100).&lt;/ref&gt;

=== Hitler ===

Fulda (2009) studied the co-evolution of newspapers and party politics in Germany, focusing primarily on Berlin, 1924-1933. During that period, the [[w:Nazi Party|Nazis (NSDAP, National Socialist German Workers' Party, Nationalsozialistische Deutsche Arbeiterpartei)]] grew from 2.6 percent of the votes for the [[w:Reichstag (Nazi Germany)|Reichstag (German parliament)]] in 1928 to 44 percent in 1933. Fulda described exaggerations in the largely tabloid press of an indecisive government incapable of managing either the economy or the increasing political violence, blamed excessively on Communists, and the potential for civil war. This turned the Nazis into an attractive choice for voters desperate for decisive action.&lt;ref&gt;Fulda (2009, Abstract, ch. 6, &quot;War of Words: The Spectre of Civil War, 1931–2&quot;.&lt;/ref&gt; 

After the 1933 elections, the Reichstag passed the &quot;[[w:Enabling Act of 1933|Enabling Act of 1933]], which gave Hitler's cabinet the right to enact laws without the consent of parliament. The Nazis then began full censorship of the newspapers, physically beating, imprisoning and in some cases killing journalists, as the leading publishers acquiesced. The primary sources of news during that period was newspapers; German radio was relatively new during that period and carried very little news. Most newspapers were [[w:Tabloid journalism|tabloids]], interested in either making money or promoting a party line with minimal regard for fact checking. A big loser in this was the right‐wing press magnate [[w:Alfred Hugenberg|Alfred Hugenberg]], whose political mismanagement led to the substantial demise of his [[w:German National People's Party|German National People's Party (Deutschnationale Volkspartei, DNVP)]], mostly benefitting Hitler.&lt;ref&gt;Fulda (2009).&lt;/ref&gt; 

This suggests the need for a [[w:Counterfactual history|counterfactual analysis of this period]], asking what kinds of changes in the structure of the media ecology might have prevented the rise of the Nazis? In particular, to what extent might a more diverse local news environment supported by citizen-directed subsidies as suggested herein have reduced the risk of a demise of democracy? And might some sort of [[w:Fairness Doctrine|fairness doctrine]] have helped?&lt;ref name=fairness&gt;Wikipedia &quot;[[w:FCC fairness doctrine|FCC fairness doctrine]]&quot;, accessed 2023-07-21.&lt;/ref&gt; And how might different rules for distributing different levels of funding to local news outlets impact the level of democratization? (Threats to democracy include legislation like the German Enabling Act of 1933 and other situations that allow an executive to successfully ignore the will of an otherwise democratic legislature, a [[w:self-coup|self-coup]], as well as a military coup.)

=== Stalin and Putin === 

[[File:Russian economic history 1885-2018.svg|thumb|Figure 7. Gross Domestic Product per person in Russian 1885-2018 in thousands of 2011 dollars]]

A 2017 survey asking Russians to name 10 of the world’s most prominent personalities listed Joe Stalin and Vladimir Putin as the top two with 38 and 34 percent, respectively. When the study was redone in 2021, Putin had slipped from number 2 to number 5. Stalin still led with 39 percent followed by Vladimir Lenin with 30 percent, Poet Alexander Pushkin and tsar Peter the Great with 23 and 19 percent each, then Putin with 15 percent.&lt;ref&gt;&lt;!-- Putin Plummets, Stalin Stays on Top in Russians’ Ranking of ‘Notable’ Historical Figures – Poll--&gt;{{cite Q|Q123197680}} &lt;!-- The most outstanding personalities in history (in Russian) --&gt;{{cite Q|Q123197317}}&lt;/ref&gt; 

It may be difficult for some people in the West to understanding how Stalin and Putin could be so popular, given the way they have been typically described in the mainstream Western media.&lt;ref&gt;[[w:Joseph Stalin|Joseph Stalin]] got much better press in the US during the Great Depression and World War II than he has gotten since 1945.&lt;/ref&gt; However, this is relatively easy to understand just by looking at the accompanying plot (Figure 7) of average annual income in that part of the world between 1885 and 2018: Both Stalin and Putin inherited economies that had fallen dramatically in the previous years and supervised dramatic improvements. Putin's decline between 2017 and 2021 may also be understood from this plot, because it shows how the dramatic growth that began around the time that Putin became acting President of Russia has slowed substantially since 2012. 

Similar comments could be made about the Vietnam war and the &quot;War on Terror&quot;.&lt;ref&gt;The Wikiversity article on &quot;[[Winning the War on Terror]]&quot; discusses the role of the media in the &quot;War on Terror&quot; and other conflicts including Vietnam.&lt;/ref&gt; To what extent can the experiments described in this article contribute to understanding the role of the media in stoking hate and fear, and how that might be impacted by citizen-direct subsidies for more and more diverse local media?

=== Iraq and the Islamic State ===

[[w:Fall of Mosul|In 2014 in Mosul, two Iraqi army divisions totaling 30,000 and another 30,000 federal police]] were overwhelmed in six days by roughly 1,500 committed Jihadists. Four months later, ''Reuters'' reported that, &quot;there were supposed to be close to 25,000 soldiers and police in the city; the reality ... was at best 10,000.&quot; Many of the missing 15,000 were &quot;ghost soldiers&quot; kicking back half their salaries to their officers. Also, &quot;[i]nfantry, armor and tanks had been shifted to Anbar, where more than 6,000 soldiers had been killed and another 12,000 had deserted.&quot;&lt;ref&gt;Parker et al. (2014).&lt;/ref&gt; 

To what extent might the political corruption and low moral documented in that ''Reuters'' report have been allowed to grow to that magnitude if Iraq had had a vigorous adversarial press, as discussed in this article? Instead, Paul Bremer, who was appointed as the [[w:Paul Bremer#Provisional coalition administrator of Iraq|Provisional coalition administrator of Iraq]] just over a week after President George W. Bush's [[w:Mission Accomplished speech|Mission Accomplished speech]] of 2003-05-01, imposed strict press censorship.&lt;ref&gt;McChesney and Nichols (2010, p. 242).&lt;/ref&gt; McChesney and Nichols contrasted this with General Eisenhower, who &quot;called in German reporters [after the official surrender of Nazi Germany in WW II] and told them he wanted a free press. If he made decisions that they disagreed with, he wanted them to say so in print.&quot;&lt;ref&gt;McChesney and Nichols (2010, Appendix II. Ike, MacArthur and the Forging of Free and Independent Press, pp. 241-254).&lt;/ref&gt;

=== Israel-Palestine ===

::''Those who make peaceful revolution impossible will make violent revolution inevitable.''
:::-- John F. Kennedy (1962)

To what extent is the [[w:Israeli–Palestinian conflict|Israeli–Palestinian conflict]] driven by differences in the media consumed by the different parties to that conflict? 

* To what extent are the supporters of Israel aware of violent acts committed by Palestinians but are ''unaware'' of the actions by Israelis that have motivated those violent acts? 
* Similarly, to what extent are the supporters of Palestinians unaware of or downplay the extent to which violence by Palestinians motivate the actions of Israel against them? 

To what extent are these differences in perceptions between supporters of Israel and supporters of Palestinians driven by differences in the media each find credible? 

What can be done to bridge this gap?  

[[w:Gene Sharp|Gene Sharp]], [[w:Mubarak Awad|Mubarak Awad]], and other supporters of [[w:nonviolence|nonviolence]] have suggested that when nonviolent direct action works, it does so by exposing a gap between the rhetoric [supported by the major media] and the reality of their opposition. Over time, this gap erodes pillars of support of the opposition. One example was the nonviolence of the [[w:First Intifada|First Intifada]] (1987-1993), which were protests against &quot;beatings, shootings, killings, house demolitions, uprooting of trees, deportations, extended imprisonments, and detentions without trial.&quot;&lt;ref&gt;Ackerman and DuVall (2000, p. 407).&lt;/ref&gt; As that campaign began, Israel got so much negative press for killing nonviolent protestors, that Israeli Defense Minister [[w:Yitzhak Rabin|Yitzhak Rabin]] ordered his soldiers NOT to kill but instead to shoot to wound. As the negative press continued, he issued wooden and metal clubs with orders to break bones.&lt;ref&gt;Shlaim (2014).&lt;/ref&gt; As the negative press still continued, Rabin ran for Prime Minister on a platform of negotiating with Palestinians. His victory and subsequent negotiations led to the [[w:Oslo Accords|Oslo Accords]] and the joint recognition of each other by the states of Israel and [[w:State of Palestine|Palestine]]. The West Bank and Gaza have continued under Israeli occupation since with some services provided by the official government of Palestine. 

During the Intifada, Israel tried to infiltrate the protestors with [[w:agent provocateur|agents provocateurs]] in Palestinian garb. They were exposed and neutralized until Israel deported 481 people they thought were leading the nonviolence who were accepted in other countries and imprisoned tens of thousands of others suspected of organizing the nonviolence. Finally, they got the violence needed to justify a massively violent repression of the Intifada.&lt;ref&gt;King (2007).&lt;/ref&gt;

The general thrust of this current analysis suggests a two pronged intervention to reduce the risk of a continuation of the violence that has marked Israel-Palestine since at least 1948: 
# Offer nonviolence training to all Palestininans, Israelis and supporters of either interested in the topic. This is the opposite of the policies Israel pursued during the First Intifada, at least according to the references cited in this discussion of that campaign.&lt;ref&gt;It also is the opposite of the decision of the US Supreme Court in ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', which ruled that teaching nonviolence to someone designated as a terrorist was a crime under the [[w:Patriot Act|Patriot Act]], as it provided &quot;material support to&quot; a foreign terrorist organization.&lt;/ref&gt;
# Provide citizen-direct subsidies to local news nonprofits in the West Bank and Gaza at, e.g., 0.15 percent of GDP, as recommended by McChesney and Nichols, cited above. 

How can we evaluate the budget required for such an experiment? The nominal GDP of the [[w:State of Palestine|State of Palestine]] in 2021 was estimated at $18 billion; 0.15% of that is $27 million. Add 10% for research to get $30 million per year. That ''annual'' cost for the media component of this proposed intervention is 12% of the billion Israeli sheckels ($246 million) that the Gaza war was costing Israel ''each day'' in the early days of the [[w:Israel-Hamas war|Israel-Hamas war]], according to the Israeli Finance Minister on 2023-10-25.&lt;ref&gt;Reuters (2023-10-25).&lt;/ref&gt; As this is being written, that war has continued for over 100 days. If the average daily cost of that war to Israel during that period has been $246 million, then that war will have already cost Israel over $24.6 billion. And that does not count the loss of lives and the destruction of property in Gaza and the West Bank. 

How much would training in nonviolence cost? That question would require more research, but if it were effective, the budget would seem to be quite modest compared to the cost of war, even if it were several times the budget for citizen-directed subsidies for local news in Palestine as just suggested.

== The decline of newspapers == 

[[File:Newspapers as a percent of US GDP.svg|thumb|Figure 8.  US newspaper revenue 1955-2020 as a percent of GDP.&lt;ref&gt;&quot;USnewspapers&quot; dataset in Croissant and Graves (2022).&lt;/ref&gt;]]

McChesney and Nichols (2022) noted that US newspaper revenue as a percent of GDP fell from over 1 percent in 1956 to less than 0.1 percent in 2020;  see Figure 8.  Abernathy (2020) noted that the US lost more than half of all newspaper journalists between 2008 and 2018.&lt;ref&gt;Abernathy (2020, p. 22).&lt;/ref&gt; A quarter of US newspapers closed between 2004 and 2020,&lt;ref&gt;Abernathy (2020, p. 21).&lt;/ref&gt; and many that still survive are publishing less, creating &quot;news deserts&quot; and &quot;ghost newspapers&quot;, some with no local journalists on staff.&lt;ref&gt;Abernathy (2020) documented the problem of increasing &quot;news deserts and ghost newspapers&quot; in the US. A local jurisdiction without a local news outlet has been called a &quot;news desert&quot;.  She uses the term &quot;ghost newspapers&quot; to describe outlets &quot;with depleted newsrooms that are only a shadow of their former selves.&quot;  Some “ghost newspapers” continue to publish with zero local journalists, produced by reporters and editors that don't live there. One example is the ''Salinas Californian'', a 125-year-old newspaper in Salinas, California, which lost its last paid journalist 2022-12, according to the Los Angeles Times (2023-03-27).  They continue to publish, though &quot;The only original content from Salinas comes in the form of paid obituaries, making death virtually the only sign of life at an institution once considered a must-read by many Salinans.&quot; A leading profiteer in this downward spiral is reportedly [[w:hedge fund|hedge fund]] [[w:Alden Global Capital|Alden Global Capital]].  Threisman (2021) reported that, &quot;When this hedge fund buys local newspapers, democracy suffers&quot;.  And Benton (2021) said, &quot;The vulture is hungry again: Alden Global Capital wants to buy a few hundred more newspapers&quot;. Hightower (2023) describes two organizations fighting this trend.  One is National Trust for Local News, a nonprofit that recently bought several local papers and &quot;is turning each publication over to local non-profit owners and helping them find ways to become sustainable.&quot; The other is CherryRoad Media, which &quot;bought 77 rural papers in 17 states, most from the predatory Gannett conglomerate that wanted to dump them&quot;, and is working to &quot;return editorial decision-making to local people and journalists ... and ... reinvest profits in real local journalism that advances democracy.&quot; News outlets acquired by something like the National Trust for Local News should be eligible for citizen-directed subsidies for local news, as discussed below, after their ownership was officially transferred to local humans.  Outlets acquired by organizations like CheeryRoad Media would not be eligible as long as they remained subsidiaries.&lt;/ref&gt; More recent news continues to be dire. The Fall 2023 issue of ''Columbia Journalism Review'' reported that 2023 &quot;has become media’s worst year on record for job losses&quot;.&lt;ref&gt;Columbia Journalism Review (2023).&lt;/ref&gt; Substantial advertising revenue has shifted to the &quot;click economy&quot;, where advertisers pay for clicks, especially on social media.&lt;ref&gt;Carter (2021).&lt;/ref&gt;

Newpapers in other parts of the world have also experienced substantial declines in revenue. In 2013 German law was changed to inclued &quot;[[w:Ancillary copyright for press publishers|Ancillary copyright for press publishers]]&quot;, also called a &quot;link tax&quot;. However, this law was declared invalid in 2019 the European Court of Justice (ECJ), because it had been submitted in advance to the [[w:European Commission|EU Commission]], as required.&lt;ref&gt;&lt;!--Axel Kannenberg (2019) ECJ: German ancillary copyright law invalid for publishers, heise online--&gt;{{cite Q|Q124051681|title= ECJ: German ancillary copyright law invalid for publishers}}&lt;/ref&gt; Before that ECJ decision, Google had removed newspapers from Google News in Germany. German publishers then reached an agreement with Google after traffic to their websites plummeted.&lt;ref&gt;&lt;!--Dominic Rushe (2014) Google News Spain to close in response to story links 'tax', Guardian--&gt;{{cite Q|Q124051847}}&lt;/ref&gt; Building on that and similar experience in Spain, the European Union adopted a [[w:Directive on Copyright in the Digital Single Market|Directive on Copyright in the Digital Single Market]] in 2019. A similar link tax proposal in Canada led [[w:Meta|Meta]], the parent company of Facebook, to withdraw news from Canada, and Google agreed to 'pay about $100 million a year into a new fund to support “news&quot;' in Canada. As of 2023-11-30, California was still considering a link tax.&lt;ref&gt;&lt;!--Ken Doctor (2023) Forget the link tax. Focus on one key metric to “save local news, NiemanLab--&gt;{{cite Q|Q124051930}}&lt;/ref&gt;

== Threats from social media == 

The growth of social media has been wonderful and terrible.  It has been wonderful in making it easier for people to maintain friendships and family ties across distances.&lt;ref&gt;Friedland (2017) noted that the Internet works well at the global level, helping people get information from any place in the world, and at the micro level, e.g., with Facebook helping people with similar diseases find one another.  It does not work well at the '“meso level arenas of communication” in the middle.  They're not big enough to aggregate all the scale that goes into creating a worldwide web or even a Wikipedia. See also Lloyd and Friedland (2016).&lt;/ref&gt; 

But it has also been terrible as  &quot;antisocial media&quot;&lt;ref&gt;Vaidhyanathan (2018).&lt;/ref&gt; have been implicated in the relatively recent rise in dysfunctional and counterproductive political polarization and violence. Ding et al. (2023) document, &quot;Same words, different meanings&quot; in their use by [[w:CNN|CNN]] and [[w:Fox News|Fox News]] and how that has interacted with word usage on Twitter between 2010 and 2020 to increase political polarization, &quot;impeding rather than supporting online democratic discourse.&quot;&lt;ref&gt;See also Ashburn.&lt;/ref&gt; 

Extreme examples of this increase have included violent efforts to prevent peaceful transitions of power in the US&lt;ref&gt;Wikipedia &quot;[[w:January 6 United States Capitol attack|January 6 United States Capitol attack]]&quot;, accessed 2023-05-09.&lt;/ref&gt; and Brazil.&lt;ref&gt;Wikipedia &quot;[[w:2023 Brazilian Congress attack|2023 Brazilian Congress attack]]&quot;, accessed 2023-05-09.&lt;/ref&gt; These changes even threaten the national security of the US and its allies,&lt;ref&gt;McMaster (2020). Zuboff (2019) noted that data on many aspects of ordinary daily life are captured and used by people with power for various purposes.  For example, data on people's locations captured from their mobile phones are used to try to sell them goods and services.  Data on a child playing with a smart Barbie doll are used to inculcate shopping habits in child and caregiver.  If you are late on a car payment, your keys can be deactivated until a tow truck can arrive to haul it away.  To what extent do the major media today have conflicts of interest in honestly reporting on this?  How might the experiments proposed herein impact the commercial calculus of major media and the political economy more generally?&lt;/ref&gt; according to [[w:H. R. McMaster|H. R. McMaster]],&lt;ref&gt;Wikipedia &quot;[[w:H. R. McMaster|H. R. McMaster]]&quot;, accessed 2023-05-09.&lt;/ref&gt; President Trump's second national security advisor. 

Various responses to these concerns have been suggested, beyond the recommendations of McChensey and Nichols.  These include the following:&lt;ref&gt;See also the section on &quot;&quot;[[International Conflict Observatory#Suggested responses to these concerns|Suggested reponses to these concerns]]&quot; in the Wikiversity article on &quot;[[International Conflict Observatory]]&quot;.&lt;/ref&gt;:

* Make internet companies liable for defamation in advertisements, similar to print media and broadcasting.&lt;ref&gt;See Baker (2020) and the Wikiversity article on &quot;[[Dean Baker on unrigging the media and the economy]], accessed 2023-07-26.&lt;/ref&gt;  
* Tax advertising revenue received by large internet companies and use that to fund more local media.&lt;ref&gt;Karr and Aaron (2019).&lt;/ref&gt; 
* Replace advertising as the source of funding for social media with subscriptions.&lt;ref&gt;Frank (2021) wrote, &quot;[D]igital aggregators like Facebook ... make money not by charging for access to content but by displaying it with finely targeted ads based on the specific types of things people have already chosen to view. If the conscious intent were to undermine social and political stability, this business model could hardly be a more effective weapon. ... [P]olicymakers’ traditional hands-off posture is no longer defensible.&quot;&lt;/ref&gt; 

To these suggestions, we add the following:  

* Allow some of but not all citizen-directed subsidies for news to go to social media outlets, as suggested below.  
* Require that all organizations whose income depends on promoting or &quot;boosting&quot; content, whether in advertisements or &quot;underwriting spots&quot; or [[w:clickbait|clickbait]], to provide copies of the ads, underwriting spots and clickbait to a central repository like the [[w:Internet Archive|Internet Archive]].  
* Use advertising to discuss overconfidence and encourage people to talk politics with humility and respect, recognizing that the primary differences they have with others are the media they consume.&lt;ref&gt;For studies of ad campaigns in other contexts, see Piwowarski et al. (2019) and Tom-Yov (2018), cited above in discussing &quot;Reducing political polarization&quot;.&lt;/ref&gt;

== How to counter political polarization ==

More research seems to be needed on how to counter the relatively recent increases in political polarization. For example, might some form of [[w:Fairness doctrine|fairness doctrine]]&lt;ref name=fairness/&gt; help reduce political polarization? [[w:Fairness doctrine#Opposition|Conservative leaders are vehemently opposed]], insisting it would be an attack on First Amendment rights. However, as noted above, the tabloid media of Germany seems to have contributed to Hitler's rise to power between 1924 and 1933. How is the increase in political polarization since 1987 and 2004 different from the disregard for fairness of the news media that helped bring Hitler to power? One example: The lawsuit ''[[w:Dominion Voting Systems v. Fox News Network|Dominion Voting Systems v. Fox News Network]]'' was settled with Fox agreeing to to pay Dominion $787.5 million while acknowledging that Fox had knowingly and intentionally made false and defamatory statements about Dominion to avoid losing audience to media outlets that continued to claim fraudulently that Donald Trump not Joe Biden had won the 2020 US presidential election. The settlement permitted Fox to avoid apologizing publicly, which could have threatened their audience share. That settlment was less than 6 percent of Fox's 2022 revenue of $14 billion.&lt;ref&gt;&lt;!--Fox earnings release for the quarter and fiscal year ended June 30, 2022--&gt;{{cite Q|Q124003735}}&lt;/ref&gt; Evidently, ''if that decision made a difference of 6 percent in their audience ratings, Fox made money from defaming Dominion even after paying them $787.5 million.'' If so, it was a good business decision, especially since they did not have to publicly apologize.  

To what extent did Fox's lies about Dominion contribute to the [[w:January 6 United States Capitol attack|mob attacks on the US Capitol on January 6, 2021]], trying to prevent the US Congress from officially declaring that Joe Biden had won the 2020 elections? And what are elected officials prepared to do to improve understanding of what contributes to increases in political polarization and how political differences can be made less lethal and more productive?

== McChesney and Nichols' Local Journalism Initiative ==

As noted above, McChesney and Nichols (2021, 2022) propose a &quot;Local Journalism Initiative&quot;, distributing 0.15 percent of GDP to local news nonprofits via local elections.  They based this partly on their earlier work suggesting that subsidies for newspapers in the US in 1840 was around 0.2 percent of GDP.&lt;ref&gt;McChesney and Nichols (2010, 2016).&lt;/ref&gt;

=== McChesney and Nichols' eligibility criteria === 

To be eligible, McChesney and Nichols say the recipient of such funds should satisfy the following:&lt;ref&gt;McChesney and Nichols (2021, 2022).  They also suggest having the US Postal Service administer this with elections every three years.&lt;/ref&gt; 
* Be a local nonprofit with at least six months of history, so voters could know their work.  
* Be locally based with at most 75 percent of salaries going to local residents.  
* Be completely independent, not a subsidiary of a larger organization.
* Produce and publish original material at least five days per week on their website for free, explicitly in the public domain. 
* Each voter is asked to vote for at least three different local news outlets to support diversity. 
* No single news outlet should get more than 25 percent of that jurisdiction's annual budget for local news subsidies.  
* Each recipient of these subsidies should get at least 1 percent of the vote to qualify, or 0.5 percent of the vote in political jurisdictions with over 1 million people. Diversity and competition are crucial.  
* There will be no content monitoring:  Government bureaucrats will not be allowed to decide what is &quot;good journalism&quot;.  That's up to the voters.  
* Voting would be limited to those 18 years and older.  

=== Alternatives === 

Some aspects of this might be relaxed for at least some political jurisdictions included in an experiment.  For example, might it be appropriate to allow for-profit news outlets to compete for these subsidies as long as they meet the other criteria?&lt;ref&gt;Kaiser (2021) noted that nonprofits in the US cannot endorse political candidates and are limited in how they can get involved in debates on political issues.  Do restrictions like these contribute to the general welfare?  Or might the public interest be better served with citizen-directed subsidies for media that might be more partisan?  This is one more question that might be answered by appropriate experimentation.&lt;/ref&gt; However, we prefer to retain the rules requiring recipients to be local and completely independent, at least for many experimental jurisdictions.&lt;ref&gt;Various contributors to Islam et al., eds. (2002) raised questions about concentrations of power in large media organizations, especially Herman, ch. 4, pp. 61-81 (73-97/336 in pdf).  Djankov et al. (2002) found that &quot;Government ownership of the media is detrimental to economic, political, and-most strikingly-social outcomes&quot;, including education and health.&lt;/ref&gt; If citizen-directed subsidies for local news go to for-profit organizations, to what extent should their finances be transparent, e.g., otherwise complying with the rules for 501(c)(3)s? Might it also be appropriate to allow some portion of these funds to be distributed to noncommercial ''social media'' outlets that submitted all their content to a public, searchable database like the [[w:Internet Archive|Internet Archive]]?   

News written by people paid with these subsidies should be available under a free license like Creative Commons Attribution-ShareAlike (CC BY-SA) 4.0 International license but not necessarily in the public domain:  Other media outlets should be free to further disseminate the news while giving credit to the organization that produced it.   

Many countries have some form of [[w:community radio|community radio]].  Some of those radio stations include what they call news and / or public affairs, and some of those are made available as podcasts via the Internet.&lt;ref&gt;In the US, many of these stations collaborate via organizations such as the [[w:National Federation of Community Broadcasters|National Federation of Community Broadcasters]], the [[w:List of Pacifica Radio stations and affiliates#Radio Stations#Affiliates|Pacifica Network Affiliates]], and the [[w:Grassroots Radio Coalition|Grassroots Radio Coalition]]. One such station with regular local news produced by volunteers in [[w:KBOO|KBOO]] in [[w:Portland, Oregon|Portland, Oregon]]; see Loving (2019).&lt;/ref&gt; If their &quot;news &amp; public affairs&quot; programs are subsequently posted to a website as podcasts, preferably accompanied by some text if not complete transcripts, under a license no more restrictive than CC BY-SA, that should make them eligible for subsidies under the criteria mentioned above if they add at least one new podcast of that nature five days per week.  If the programming of this nature that they produce is ''not'' available on the web or under an appropriate license, part of any experiments as discussed here might include offers to help such radio stations become eligible.  

Might it be wise to allow children to vote for news organizations they like?  Ryan Sorrell, founder and publisher of the Kansas City Defender, insists that, &quot;young people ... are very interested in news. It just has to be produced and packaged the right way for them to be interested in consuming it&quot;.&lt;ref&gt;Holmes (2022).&lt;/ref&gt; The French-language [[:fr:w:Topo (revue)|''Topo'']] present news and complex issues in comic strip format. Their co-editor in chief insists, &quot;there are plenty of ways to get young people interested in current affairs&quot;.&lt;ref&gt;Biehlmann (2023).&lt;/ref&gt; Might allowing children to vote for news outlets they like increase public interest in learning and in civic engagement among both children and their caregivers?  Should this be tested in some experimental jurisdictions?&lt;ref&gt;We may not want infants who cannot read a simple children's book to vote for &quot;news&quot;, but if they can read the names of eligible local news outlets on a ballot, why not encourage them to vote?  As Rourmeen Islam wrote in 2002, &quot;erring on the side of more freedom rather than less would appear to cause less harm.&quot; (World Bank, 2002, pp. 21-22; 33-34/336 in pdf).&lt;/ref&gt;

Some of the money may go to media outlets that seem wacko to many voters.  However, how different might that be from the current situation? Most importantly, if these subsidies have the effect that Tocqueville reported from 1831, they should be good for democracy and for broadly shared peace and prosperity for the long term:  They could stimulate public debate, and wacko media might have ''less'' power than they currently do, with &quot;each separate journal exercis[ing] but little authority; but the power of the periodical press [being] second only to that of the people.&quot;&lt;ref&gt;Tocqueville (1835; 2001, p. 94).&lt;/ref&gt; 

Tocqueville's comparison of newspapers in France and the US in 1831 is echoed in Cagé's (2022) concern about &quot;the Fox News effect&quot; in the US and that of Bolloré in France.  She cites research claiming that biases in Fox News made major contributions to electing Republicans in the US since 2000.&lt;ref&gt;Cagé (2022, pp. 21-22, 59-60).  She cited DellaVigna and Ethan Kaplan (2007), who reported that Fox News had introduced cable programming into 20 percent of town in the US between 1996 and 2000.  They found that the presence of Fox increased the vote share for Republicans between 0.4 and 0.7 percentage points over neighboring non-Fox towns that seemed otherwise indistinguishable.  In 2000 Fox News was available in roughly 35 percent of households, which suggests that Fox News shifted the nationwide vote tally by between 0.15 and 0.2 percentage points.  They conclude that this shift was small but likely decisive in the close 2000 US presidential election.&lt;/ref&gt; These shifts, including changes by the conservative-leaning broadcasting company, Sinclair Broadcast Group, reportedly made a substantive contribution to the election of [[w:Donald Trump|Donald Trump]] as US President in 2016, while a comparable estimate of the impact of changes in MSNBC &quot;is an imprecise zero.&quot;&lt;ref&gt;Cagé (2022, pp. 21-22).  Miho (2022) analyzes the timing of the introduction of biased programming by the conservative-leaning broadcasting company, Sinclair Broadcast Group, between 1992 and 2020, comparing counties in the US with and without a Sinclair station.  This work estimates a 2.5 percentage point increase in the Republican vote share during the 2012 US presidential election and double that during the 2016 and 2020 presidential elections with comparable increases in Republican representation in the US Congress.&lt;/ref&gt; In France, she provides documentation claiming that the media empire of French billionaire [[w:Vincent Bolloré|Vincent Bolloré]] has made a major contribution to the rise of far-right politician [[w:Éric Zemmour|Éric Zemmour]] and is buying media in Spain.&lt;ref&gt;Cagé (2022, pp. 24, 60)&lt;/ref&gt; The pattern is simple:  Fire journalists and replace them with talk shows, which are cheaper to produce and are popular, evidently exploiting the [[w:overconfidence effect|overconfidence effect]].  

To what extent is the increase in political polarization since 1987&lt;ref name=fairness/&gt; and 2004&lt;ref&gt;Wikipedia &quot;[[w:Facebook|Facebook]]&quot;, accessed 2023-07-21.&lt;/ref&gt; due to increased concentration of ownership of both traditional and social media (and how those organizations make money selling changes in audience behaviors to the people who give them money)? To the extent that this increase in polarization has been driven by those changes in the media, citizen-directed subsidies for diverse news should reverse that trend. This hypothesis can be tested by experiments like those proposed herein.

== Roadmap for local news == 

Green et al. (2023) describe &quot;an emerging approach to meeting civic information needs&quot; in a &quot;Roadmap for local news&quot;. This report insists that society needs &quot;civic information&quot;, not merely &quot;news&quot;. It summarizes interviews with 51 leaders from nonprofit and commercial media across all forms of distribution (print, radio, broadcast, digital, SMS) in member organizations, news networks, news funders and researchers. They say that, &quot;Rampant disinformation is being weaponized by extremists&quot;, and &quot;Democratic participation and representation are under threat.&quot; They recommend four strategies to address &quot;this escalating information crisis&quot;: 

# Coordinate work around the goal of expanding “civic information,” not saving the news business;
# Directly invest in the production of civic information;
# Invest in shared services to sustain new and emerging civic information networks; and
# Cultivate and pass public policies that support the expansion of civic information while maintaining editorial independence. 

Part of the motivation for this article on &quot;Information is a public good&quot; is the belief that solid research on the value of such interventions should both (a) make it easier to get the funding needed, and (b) help direct the funding to interventions that seem to make the maximum contributions to improving broadly shared peace and prosperity for the long term at minimum cost.

== Budgets for experiments == 

What factors should be considered in evaluating budgets for experiments to estimate the impact of citizen-directed subsidies for news?   

[[File:Advertising as a percent of Gross Domestic Product in the United States.svg|thumb|Figure 9.  Advertising as a percent of Gross Domestic Product in the United States, 1919 to 2007.&lt;ref name=ads&gt;Galbi (2008).&lt;/ref&gt;]]  

Rolnik et al. (2019) suggested that $50 per person, roughly 0.08 percent of US GDP, might be enough.  However, that's a pittance compared to the revenue lost by newspapers in the US since 1955, as documented in Figure 8 above.  It's also a pittance compared to the money spent on advertising (see Figure 9):  Can we really expect local media funded with only 0.08 or 0.15 percent of GDP to compete with media funded by 2 percent of GDP?  Maybe, but that's far from obvious.  Might it be prudent to fund local journalism in some experimental jurisdictions at levels exceeding the money spent on advertising, i.e., at roughly 2 percent of GDP or more?  If information is a public good, as suggested by the research summarized here, then such high subsidies would be needed in some experimental jurisdictions, because the maximum of anything (including net benefits = benefits minus costs) cannot be confidently identified without conducting some experiments ''beyond the point of diminishing returns''.&lt;ref&gt;A parabola can be estimated from three distinct points.  However, in fitting a parabola or any other mathematical model to empirical data, one can never know if an empirical phenomenon has been adequately modeled and a maximum adequately estimated without data near the maximum and on both sides of it (unless the maximum is at a boundary, e.g., 0).  See, e.g., Box and Draper (2007).&lt;/ref&gt;  

[[File:AccountantsAuditorsUS.svg|thumb|Figure 10.  Accountants and auditors as a percent of the US workforce.&lt;ref name=actg&gt;Accountants and auditors as a percent of US households, 1850 - 2016, using the OCC1950 occupation codes in a sample of households available from from the [[w:IPUMS|Integrated Public Use Microdata Series at the University of Minnesota (IPUMS)]].  For more detail see the &quot;AccountantsAuditorsPct&quot; data set in the &quot;Ecdat&quot; package and the &quot;AccountantsAuditorsPct&quot; vignette in the &quot;Ecfun&quot; package available from within the [[w:R (programming language)|R (programming language)]] using 'install.packages(c(&quot;Ecdat&quot;, &quot;Ecfun&quot;), repos=&quot;http://R-Forge.R-project.org&quot;)'.&lt;/ref&gt;]] 

Also, news might serve a roughly comparable function to accounting and auditing, as both help reduce losses due to incompetence, malfeasance and fraud.  Two points on this:  

# CONTROL FRAUDS:  Black (2013) noted that many heads of organizations can find accountants and auditors willing to certify accounting reports they know to be fraudulent. Black calls such executives &quot;control frauds.&quot;&lt;ref&gt;Black (2013). &lt;/ref&gt; Primary protections against these kinds of problems are vigorous, independent journalists and more money spent on independent evaluations beyond the control of such executives.  In this regard, we note two major differences between the [[w:Savings and loan crisis|Savings &amp; Loan scandal]] of the late 1980s and early 1990s&lt;ref&gt;Wikipedia &quot;[[w:Savings and loan crisis|Savings and loan crisis]]&quot;, accessed 2023-06-25.&lt;/ref&gt; and the [[w:2007–008 financial crisis|international financial meltdown of 2007-2008]]:&lt;ref&gt;Wikipedia &quot;[[w:2007–008 financial crisis|2007–008 financial crisis]]&quot;, accessed 2023-06-25.&lt;/ref&gt; First the major banks by 2007 were much bigger and controlled much larger advertising budgets than the Saving &amp; Loan industry did 15-20 years earlier.  This meant that major media had a much bigger conflict of interest in honestly reporting on questionable activities of these major accounts.  Second, the major banks had made much larger political campaign contributions to much larger portions of both the US House and Senate. However, might the massive amounts of big money spent on campaign finance have been as effective if the major media did not have such a conflict of interest in exposing more details of the corrosive impact of major campaign donors on the quality of government?  To what extent might this corrosive impact be quantified in experimental polities?  
# ADEQUATE RESEARCH OF OUTCOMES:  Many nonprofits and governmental agencies officially have outcome measures, but many of those measures tend to be relatively superficial like the number of people served.  It's much harder to evaluate the actual benefits to the people served and to society.  For example, the Perry Preschool&lt;ref&gt;Schweinhart et al. (2005).  See also Wikipedia &quot;[[w:HighScope|HighScope]]&quot;, accessed 2023-06-15. &lt;/ref&gt; and Abecedarian&lt;ref&gt;e.g., Sparling and Meunier (2019).  See also Wikipedia &quot;Abecedarian Early Intervention Project&quot;, accessed 2023-06-25.&lt;/ref&gt; programs divided poor children and caregivers into experimental and control groups and followed them for decades to establish that their interventions were enormously effective.&lt;ref&gt;For more recent research on the economic value of high quality programs for early childhood development, see, e.g., &lt;!-- &quot;The Heckman Equation&quot; website (heckmanequation.org)--&gt;{{cite Q|Q121010808}}, accessed 2023-07-29.&lt;/ref&gt; Meanwhile, US President Lyndon Johnson's [[w:Great Society|Great Society]] programs,&lt;ref&gt;Wikipedia &quot;[[w:Great Society|Great Society]]&quot;, accessed 2023-07-11.&lt;/ref&gt; and [[w:Head Start|Head Start]] in particular, did not invest as heavily in research.  That lack of documentation of results made them a relatively easy target for political opponents claiming that government is the problem, not the solution. These counter arguments were popularized by US President Ronald Reagan and UK Prime Minister Margaret Thatcher to justify reducing or eliminating government funds for many such programs. Banerjee and Duflo (2019) summarized relevant research in this area by saying that the programs were not the disaster that Reagan, Thatcher, and others claimed, but they were also not as efficient and effective as they could have been, because many local implementations were underfunded, poorly managed and poorly evaluated.   Bedasso (2021) analyzed World Bank projects completed from 2009 to 2020, concluding that high quality monitoring and evaluation on average made a major contribution to the positive results from the successful projects studied.&lt;ref&gt;See also Raimondo (2016).&lt;/ref&gt; To what extent might citizen-directed subsidies for local media as suggested here improve the demand for (and the supply of) better evaluations, leading to better programs and the lower crime, etc., that came from those programs?  To what extent might this effect be quantified using randomized controlled trials comparing different jurisdictions, analogous to the research for which Banerjee, Duflo, and Kramer won the 2019 Nobel Memorial Prize in Economics?  

This discussion makes us wonder if better research and better news might deliver dramatically more benefits than costs in reducing money wasted on both funding wasteful programs and on failing to fund effective ones?  In particular, might society benefit from matching the 1 percent of the workforce occupied by accountants and auditors with better research and citizen-directed subsidies for news (see Figure 10)?  If, for example, 1 or 2 percent of GDP distributed to local news nonprofits via local elections, as described above, increased the average rate of economic growth in GDP per capita by 0.1 percentage point per year, that increase would accumulate over time, so that after 10 or 20 years, the news would in effect become free, paid by money that implementing political jurisdictions would not have without those subsidies. Moreover those accumulations might remain as long as they were not wiped out by events comparable to the economic disasters documented above in discussing &quot;Stalin and Putin&quot; -- and maybe not even then as suggested by Figure 7. 

=== Other recommendations and natural experiments === 

Table 1 compares the recommendations of McChesney and Nichols (2021, 2022) and Rolnik et al. (2019) with other possible points of reference. Crudely similar to McChesney, Nichols, and Rolnik et al., Karr (2019) and Karr and Aaron (2019) recommend &quot;a 2 percent ad tax on all online enterprises that in 2018 earned more than $200 million in annual digital-ad revenues&quot;. They claim that this &quot;would yield more than $1.8 billion a year&quot;,  which is very roughly 0.008 percent of GDP, $5 per person per year;&lt;ref name=Karr&gt;Karr (2019), Karr and Aaron (2019). US GDP for 2019 was $21,381 billion, per International Monetary Fund (2023). Thus, $1.8 billion is 0.0084% of US GDP and $5.44 for each of the 330,513,000 humans in the US in 2019; round to 0.008% and $5 per capita.&lt;/ref&gt; Google has negotiated agreements similar to this with the governments of Australia and Canada.&lt;ref&gt;Hermida (2023).&lt;/ref&gt; Other points of reference include the percent of GDP devoted to accounting and auditing and advertising. As displayed in Figure 10, accountants and auditors are roughly 1 percent of the workforce in the US. It's not clear how to translate that into a percent of GDP, but 2 percent seems like a reasonable approximation, if we assume that average income of accountants and auditors is a little above the national norm and overhead is not quite double their salaries; this may be conservative, because many accountants and auditors have support staff, who are not accountants nor auditors but support their work. 

Another point of reference is the average annual growth rate in GDP per capita since World War II: A subsidy of 2 percent of GDP would be roughly one year's increase in average annual income since World War II, as noted with Figure 1 above. More precisely, the US economy (GDP per capita adjusted for inflation) was 2.2 percent per year between between 1950 and 1990 but only 1.1 percent between 1990 and 2022, according to inequality expert [[w:Thomas Piketty|Thomas Piketty]], who attributed that slowing in the rate of economic growth to the increase in income inequality in the US since 1975, documented in Fgure 6 above. Whether Piketty is correct or not, if 2 percent per year subsidies for journalism close the gap between 1.1 and 2.2 percent per year, those media subsidies would effectively become free after two years, paid out of income the US would not have without them. This reinforces the main point of this essay regarding the need for randomized controlled trials on any intervention with a credible claim to improving the prospects for broadly shared economic growth for the long term.&lt;ref name=GDP&gt;The growth in US GDP per capita is discussed in the working paper on Wikiversity titled, &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;, accessed 2021-05-19.   For a similar comment about an intervention that increased the rate of economic growth becoming free, paid out of income we would not have without it, has been made about the impact of improving education by {{cite Q|Q56849246}}&lt;!-- Endangering Prosperity: A Global View of the American School--&gt;, p. 12.&lt;/ref&gt;    
This table includes other interventions for which humanity would benefit from more substantive evaluation of their impact. This includes [[w:Democracy voucher|Seattle's &quot;Democracy Voucher&quot; program]], which gives each registered voter four $25 vouchers, totaling $100, which they could give to eligible candidates running for municipal office. However, only the first 47,000 were honored;  this limited the city's commitment to $4.7 million every other year.&lt;ref name=Berman&gt;Berman (2015). The Wikipedia article on [[w:Seattle]] says that the gross metropolitan product (GMP) for the Seattle-Tacoma metropolitan area was $231 billion in 2010 for a population of 3,979,845.  That makes the GMP per capita roughly $58,000.  However, the population of Seattle proper was only 608,660 in 2010, making the Gross City Product roughly $35 billion.  $4.7 million is 0.0133 percent of $35 billion.  However, that's very other year, so it's really only 0.007 percent of the Gross City Product.&lt;/ref&gt; If Seattle can afford $100 per registered voter, many other governmental entities can afford something very roughly comparable for each adult in their jurisdiction. Seattle's &quot;democracy vouchers&quot; are used to fund political campaigns, not local media; they are mentioned here as a point of comparison. 

Other interventions that seem to deserve more research than we've seen are the [[w:New Jersey Civic Information Consortium|New Jersey Civic Information Consortium]] (NJCIC)&lt;ref name=njcicBudget&gt;Karr (2020). Per [[w:New Jersey]] the Gross State Product in 2018 was roughly $640 billion;  it's population in 2020 was roughly 9.3 million. The initial  $500,000 for the project is only $0.05 per person per year and only 0.00008 percent of $640 billion.&lt;/ref&gt; and a program in California to improve local news in communities in dire need of strong local journalism. The NJCIC was initially funded at $500,000, which is only 0.00008 percent of New Jersey economy (GDP) of $640 billion. In 2022, the state of California authorized $25 million for up to 40 Berkeley local news fellowships offering &quot;a $50,000 annual stipend [for 3 years] to supplement their salaries while they work in California newsrooms covering communities in dire need of strong local journalism.&quot; This Berkeley program is roughly $0.21 per person per year, 0.0007 percent of the Gross State Produce of $3.6 trillion that year, for an annual rate of very roughly 0.0002 percent of the Gross State Product.&lt;ref name=Berkeley&gt;Natividad (2023) discusses the Berkeley local news fellowships. California Gross State Product from US Bureau of Economic Analysis (2023). California population on 2022-07-01 from US Census Bureau (2023).&lt;/ref&gt; A similar project in Indiana funded by philanthropies began as the Indiana Local News Initiative&lt;ref&gt;Greenwell (2023).&lt;/ref&gt; and has morphed into Free Press Indiana.&lt;ref&gt;See &quot;[https://www.localnewsforindiana.org LocalNewsForIndiana.org]&quot;; accessed 29 December 2023.&lt;/ref&gt;

Some local [[w:League of Women Voters|Leagues of Women Voters]] have all-volunteer teams who observe official meetings of local governmental bodies and write reports.&lt;ref&gt;Wilson (2007).&lt;/ref&gt; The [[w:City Bureau|City Bureau]] nonprofit news organization in Chicago, Illinous, &quot;trains and pays community members to attend local government meetings and report back on them.&quot;&lt;ref&gt;See &quot;[https://www.citybureau.org/documenters-about citybureau.org/documenters-about]&quot;.&lt;/ref&gt; The program has been so successful, it has exanded to other cities.&lt;ref&gt;Greenwell (2023).&lt;/ref&gt;

For an international comparison, we include [[w:amaBhungane|amaBhungane]],&lt;ref name=amaBhu&gt;The budget for [[w:AmaBhungane#Budget|amaBhungane]] in 2020 was estimated at 590,000 US dollars at the current exchange rate, per analysis in the [[w:AmaBhungane#Budget|budget]] section of the Wikipedia article on amaBhungane. That's 0.00017 percent of South African's nominal GDP for that year of 337.5 million US dollars, per the section on &quot;[[w:Economy of South Africa#Historical statistics 1980–2022|Historical statistics 1980–2022]]&quot; in the Wikipedia article on [[w:Economy of South Africa|Economy of South Africa]]; round that to 0.002 percent for convenience. The population of South Africa that year was estimated at 59,309,000, according to the section on &quot;[[w:Demographics of South Africa#UN Age and population estimates: 1950 to 2030|UN Age and population estimates: 1950 to 2030]]&quot; in the Wikipedia article on [[w:Demographics of South Africa|Demographics of South Africa]]; this gives a budget of 1 penny US per capita. (All these Wikipedia articles were accessed 2023-12-28.)&lt;/ref&gt; whose investigative journalism exposed a corruption scandal that helped force South African President [[w:Jacob Zuma|Jacob Zuma]] to resign in 2018; amaBhungane's budget is very roughly one penny US per person per year in South Africa, 0.0002 percent of GDP. To the extent that this essay provides a fair and balanced account of the impact of journalism on political economy, South African and the rest of the world would benefit from more funding for amaBhungane and other comparable investigative journalism organizations. This could initially include randomized controlled trials involving citizen-directed subsidies for local news outlets in poor communities in South Africa and elsewhere, as we discuss further in the rest of this essay. Without such experiments, we are asking for funds based more on faith than science. 

{| class=&quot;wikitable sortable&quot; 
!option / reference
!% of GDP 
!colspan=2|US$
!per …
!
|-
|style=&quot;text-align:left;&quot;|US postal subsidies for newspapers 1840-44 
| 0.21%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $140
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
| &lt;ref name=McC-N2010/&gt;
|-
|style=&quot;text-align:left;&quot;|McChesney &amp; Nichols (2021, 2022)
| 0.15%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $100
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
| &lt;ref name=McC-N2021/&gt;
|-
|[[Confirmation bias and conflict#Relevant research|Rolnik et al.]] 
| 0.08%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $50
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|adult &amp; year
| &lt;ref name=Rolnik/&gt; 
|-
|[[w:Free Press (organization)|Free Press]]
| 0.008%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $6
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref name=Karr/&gt;
|-
|[[w:Democracy vouchers|Democracy vouchers]]
| 0.007% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $100
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|voter &amp; municipal election for the first 47,000
|&lt;ref name=Berman/&gt;
|-
|[[Confirmation bias and conflict#Advertising and accounting|advertising]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref name=ads/&gt;
|-
|[[Confirmation bias and conflict#Advertising and accounting|accounting]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year
|&lt;ref&gt;As noted with Figure 10 and the discussion above, accountants and auditors are roughly 1 percent of the US workforce, and it seems reasonable to guess that their pay combined with support staff and overhead would likely make them roughly double that, 2 percent, as a portion of GDP.&lt;/ref&gt;
|-
|[[US Gross Domestic Product (GDP) per capita|US productivity improvements]]
| 2% 
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $1,300
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .00
|person &amp; year (GDP per capita)
|&lt;ref&gt;For an analysis of the rate of growth in US GDP per capita, see the working paper on Wikiversity titled, &quot;[[US Gross Domestic Product (GDP) per capita]]&quot;&lt;/ref&gt;
|-
|[[w:New Jersey Civic Information Consortium|New Jersey Civic Information Consortium]]
| 0.00008%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .05
|person &amp; year
|&lt;ref name=njcicBudget/&gt;
|-
| Berkeley local news fellowships  
| 0.0002%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .21
|person &amp; year
|&lt;ref name=Berkeley/&gt;
|-
|[[w:amaBhungane|amaBhungane]]
| 0.0002%
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | $0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .01
|person &amp; year in South Africa
|&lt;ref name=amaBhu/&gt;
|}
Table 1. Media subsidies and other points of reference. At the low end, political corruption exposed in part by amaBhungane forced the resignation in 2018 of South African President Zuma on a budget that's very roughly one penny US per person per year. If much higher subsidies of 1 percent of GDP restored an annual growth rate of 2.2 percent per year from the more recent 1.1 percent discussed with Figure 1 above, those subsidies would pay for themselves from one year's growth that the US would not otherwise have.

== Other factors == 

We feel a need here to suggest other issues to consider in designing experiments to improve the political economy:  education, empowering women, free speech, free press, peaceful assembly, and reducing political polarization.

EDUCATION:  Modern research suggests that society might have lower crime&lt;ref&gt;Wang et al. (2022).&lt;/ref&gt; and faster rates of economic growth with better funding for and better research&lt;ref&gt;Hanushek and Woessmann (2015).&lt;/ref&gt; on quality child care from pregnancy through age 17. 

EMPOWERING WOMEN:  Might the best known way to limit and reverse population growth be to empower women and girls?  Without that, might the human population continue to grow until some major disaster reduces that population dramatically?&lt;ref&gt;Roser (2017).&lt;/ref&gt;

FREE SPEECH, FREE PRESS, PEACEFUL ASSEMBLY:  Verbitsky said, &quot;Journalism is disseminating information that someone does not want known; the rest is propaganda.&quot;&lt;ref&gt;Verbitsky (2006, p. 16), author's translation from Spanish.&lt;/ref&gt; This discussion of threats, arrests, kidnappings, and murders of journalists&lt;ref&gt;Monitored especially by the [[w:Committee to Protect Journalists|Committee to Protect Journalists]], as discussed in the Wikipedia article on them, accessed 2023-07-04.&lt;/ref&gt; and violent suppression of peaceful assemblies&lt;ref&gt;Monitored by Freedom House and others.  See, e.g., the Wikipedia article on &quot;[[w:List of freedom indices|List of freedom indices]]&quot;, accessed 2023-07-04.&lt;/ref&gt; encourages us to consider the potential utility of efforts to improve local news, as noted by contributors to Islam et al., eds. (2002), cited above.  Data on such problems should be considered in selecting sites for experiments with citizen-directed subsidies for journalism and in analyzing the results from such experiments. Such data should include the incidence of legal proceedings against journalists and publishers&lt;ref&gt;Including the risks of [[w:trategic lawsuit against public participation|strategic lawsuits against public participation]] (SLAPPs) and other questionable uses of the courts including some documented in the &quot;[[w:Freedom of the Press Foundation#U.S. Press Freedom Tracker|U.S. Press Freedom Tracker]]&quot;, mentioned above.&lt;/ref&gt; as well as threats, murders, etc., in jurisdictions comparable to experimental jurisdictions.  

REDUCING POLITICAL POLARIZATION:  What interventions might be tested that would attempt to reduce political polarization while also experimenting with increasing funding for news through small, diverse news organizations? For example, might an ad campaign feature someone saying, &quot;We don't talk politics&quot;, with a  reply, &quot;We have to talk politics with humility and mutual respect, because the alternative is killing people over misunderstandings&quot;? Might another ad say, &quot;Don't get angry: Get curious&quot;?  What can be done to encourage people to get curious rather than angry when they hear something that contradicts their preconceptions?  How can people be encouraged to talk politics with humility and respect for others, understanding that everyone can be misinformed and others might have useful information?&lt;ref&gt;Wikiversity &quot;[[How can we know?]]&quot;, accessed 2023-07-22, reviews relevant research relating to political polarization.  Yom-Tov et al. (2018) described a randomized-controlled trial that compared the effectiveness of different advertisements &quot;to improve food choices and integrate exercise into daily activities of internet users.&quot; They found &quot;powerful ways to measure and improve the effectiveness of online public health interventions&quot; and showed &quot;that corporations that use these sophisticated tools to promote unhealthy products can potentially be outbid and outmaneuvered.&quot; Similar research might attempt to promote strategies for countering political polarization.  See also Piwowarski  et al. (2019).&lt;/ref&gt;  

FOCUS ON POLITICIANS: Mansuri et al. (2023) randomly assigned presidents of village governments in the state of [[w:Tamil Nadu|Tamil Nadu]] in India to one of three groups with (1) a financial incentive or (2) a certificate with an information campaign (without a financial incentive) for better government or (3) a control group. They found that the public benefitted from both the financial and non-financial incentives, and the non-financial incentives were more cost effective. Might it make sense in some experimental jurisdictions to structure the subsidies for local news by asking voters to allocate, e.g., half their votes for local news to outlet(s) that they think provide the best information about politicians with the other half based on &quot;general news&quot;?&lt;ref&gt;Mansuri et al. (2023).&lt;/ref&gt;

PIGGYBACK ON COMMUNITY AND LOCAL DEVELOPMENT PROGRAMS:  The World Bank (2023) notes that, &quot;Experience has shown that when given clear and transparent rules, access to information, and appropriate technical and financial support, communities can effectively organize to identify community priorities and address local development challenges by working in partnership with local governments and other institutions to build small-scale infrastructure, deliver basic services and enhance livelihoods. The World Bank recognizes that CLD [Community and Local Development]  approaches and actions are important elements of an effective poverty-reduction and sustainable development strategy.&quot; This suggests that experiments in citizen-directed subsidies for news might best be implemented as adjuncts to other CLD projects to improve &quot;access to information&quot; needed for the success of that and follow-on projects.  Such news subsidies should complement and reinforce the quality of monitoring and evaluation, which was &quot;significantly and positively associated with project outcome as institutionally measured at the World Bank&quot;.&lt;ref&gt;Raimondo (2016). See also Bedasso (2021).&lt;/ref&gt;

Simple experiments of the type suggested here might add citizen-directed subsidies for local news to a random selection of Community and Local Development (CLD) projects.  Such experimental jurisdictions should be small enough that the budget for the proposed citizen-directed subsidies for news would be seen as feasible but large enough so appropriate data could be obtained and compared with control jurisdictions not receiving such subsidies for local news.

However, some potential recipients of CLD funding may be in news deserts or with &quot;ghost newspapers&quot;, as mentioned above.  Some may not have at least three local news outlets that have been publishing something they call news each workday for at least six months, as required for the local elections recommended by McChesney and Nichols (2021, 2022), outlined above.  In such jurisdictions, the local consultations that identify community priorities for CLD funding should also include discussions of how to grow competitive local news outlets to help the community maximize the benefits they get from the project.  

The need for &quot;at least three local news outlets&quot; is reinforced by the possibilities that two or three local news outlets may be an [[w:oligopoly|oligopoly]], acting like a monopoly.  This risk may be minimized by working to ''limit'' barriers to entry and to encourage different news outlets to serve different segments of the market for news.  The risks of oligopolistic behavior may be further reduced by requiring all recipients of citizen-directed subsidies to release their content under a free license like the Creative Commons Attribution-ShareAlike (SS BY-SA) 4.0 international license.  This could push each independent local news outlet to spend part of their time reading each other's work while pursuing their own journalistic investigations, hoping for scoops that could attract a wider audience after being cited by other outlets.&lt;ref&gt;Wikipedia &quot;[[w:Oligopoly|Oligopoly]]&quot;, accessed 2023-07-06.&lt;/ref&gt;

This preference for at least three independent local news outlets in an experimental jurisdiction puts a lower bound on the size of jurisdictions to be included as experimental units, especially if we assume that the independent outlets should employ on average at least two journalists, giving a minimum of six journalists employed by local news outlets in an experimental jurisdiction.  The discussions above suggested subsidies ranging from 0.08 percent to 2 percent or more.  To get a lower bound for the size of experimental jurisdictions, we divide 6 by 0.08 and 2 percent:  Six journalists would be 0.08 percent of a population of 7,500 and 2 percent of a population of 300.

== Sampling units / experimental polities == 

Many local governments could fund local news nonprofits at 0.15% of GDP, because it would likely be comparable to what they currently spend on accounting, media and public relations.&lt;ref&gt;&quot;State and local governments [in the US] spent $3.5 trillion on direct general government expenditures in fiscal year 2020&quot;, with states spending $1.7 trillion and local governments $1.8 trillion&quot;, per Urban Institute (2024). The nominal GDP of the US for 2020 was $21 trillion, per International Monetary Fund (2024). Thus, local government spending $1.7 trillion is 1.8% of the $21 trillion US GDP, which is comparable to the money spent on accounting per Figure 10 and advertising per Figure 9.&lt;/ref&gt; If the results of such funding are even a modest percent of the benefits claimed in the documents cited above, any jurisdiction that does that would likely obtain a handsome return on that investment. 

Jurisdictions for randomized controlled trials might include some of the members of the United Nations with the smallest Gross Domestic Products (GDPs) or even some of the poorest census-designated places&lt;ref&gt;Wikipedia &quot;[[w:Census-designated place|Census-designated place]]&quot;, accessed 2023-07-11.&lt;/ref&gt; in a country like the US. Alternatively, they might include areas with seemingly intractable cycles of violence like Israel and Palestine: The budget for interventions like those proposed herein are a fraction of what is being spent on defense and on violence challenging existing power structures. If interventions roughly comparable to those discussed herein can reduce the lethality of a conflict at a modest cost, it would have an incredible return on investment (ROI). That would be true not merely for the focus of the intervention but for other similar conflicts. 

For illustration purposes only, Table 2 lists the six countries in the United Nations with the smallest GDPs in 2021 in US dollars at current prices according to the United Nations Statistics Division plus Palestine and Israel, along with their populations and GDP per capita plus the money required to fund citizen-directed subsidies at 0.15 percent of GDP, as recommended by McChesney and Nichols (2021, 2022).  The rough budgets suggested here would only be for a news subsidy companion to Community and Local Development (CLD) projects, as discussed above or for intervention(s) attempting to reduce the lethality of conflict.  Other factors should be considered in detailed planning.  For example, the budget for such a project in [[w:Montserrat|Montserrat]] may need to be increased to support greater diversity in the local news outlets actually subsidized.  And a careful study of local culture in [[w:Kiribati|Kiribati]] may indicate that the suggested budget figure there may support substantially fewer than the 97 journalists suggested by the naive computations in this table.  The key point, however, is that subsidies of this magnitude would be modest as a proportion of many other projects funded by agencies like the World Bank or the money spent on defense or war. 

{| class=&quot;wikitable sortable&quot; style=text-align:right
! Country !! Population !! GDP / capita
! GDP (million USD) 
! annual subsidy at 0.15% of GDP ($K) 
! number of journalists&lt;sup&gt;(*)&lt;/sup&gt;
|-
| [[w:Tuvalu|Tuvalu]] || 11,204 || $5,370 || $60 || $90 ||8.4
|-
| [[w:Montserrat|Montserrat]] || 4,417 || $16,199 || $72 || $107 || 3.3
|- 
| [[w:Nauru|Nauru]] || 12,511 || $12,390 || $155 || $233 || 9.4
|- 
| [[w:Palau|Palau]] || 18,024 || $12,084 || $218 || $327 || 13.5
|- 
| [[w:Kiribati|Kiribati]] || 128,874 || $1,765 || $227 || $341 ||96.7
|- 
| [[w:Marshall Islands|Marshall Islands]] || 42,040 || $6,111 || $257 || $385 || 31.5
|- 
| [[w:State of Palestine|State of Palestine]] || 5,483,450 || $3,302 || $18,037 || $27,055 || 4,113
|- 
| [[w:Israel| Israel]] || 9,877,280 || $48,757 || $481,591 || $722,387 || 7,408 
|}
Table 2.  Rough estimate of the budget for subsidies at 0.15 percent of GDP for the 6 smallest members of the UN plus Palestine and Israel. Population and GDP at current prices per United Nations Statistics Division (2023). (*) &quot;Number of journalists&quot; was computed assuming each journalist would cost twice the GDP / capita.  For example, the GDP / capita for Tuvalu in this table is $5,370. Double that to get $10,740.  Divide that into $90,000 to get 8.4.  

Other possibilities for experimental units might be historically impoverished subnational groups like [[w:Native Americans in the United States|Native American jurisdictions in the United States]].  As of 12 January 2023 there were &quot;574 Tribal entities recognized by and eligible for funding and services from the [[w:Bureau of Indian Affairs|Bureau of Indian Affairs]] (BIA)&quot;, some of which have multiple subunits, e.g., populations in different counties or census-designated places. For example, the largest is the [[w:Navajo Nation|Navajo Nation Reservation]] that is split between Arizona, New Mexico, and Utah.&lt;ref&gt;Newland (2023).&lt;/ref&gt; Some of these subdivisions are too small to be suitable for experiments in citizen-directed subsidies for news.  Others have subdivisions large enough so that some subdivisions might be in experimental group(s) with others as controls.  If there are at least three subdivisions with sufficient populations, at least one could be a control, with others being Community and Local Development (CLD) projects both with and without companion news subsidies, as discussed above.&lt;ref&gt;Data analysis might consider spatial autocorrelation, as used by Mohammadi et al. (2022) and multi-level time series text analysis, used by Friedland et al. (2022).  The latter discuss &quot;Asymmetric communication ecologies and the erosion of civil society in Wisconsin&quot;:  That state had historically been moderate &quot;with a strong progressive legacy&quot;.  Then in 2010 they elected a governor who attacked the state's public sector unions with substantial success and voted for Donald Trump for President in 2016 but ''against'' him in 2020.&lt;/ref&gt;

== Supplement not replace other funding == 

The subsidies proposed here should supplement not replace other funding, similar to the subsidies under the US Postal Service Act of 1792.  McChesney and Nichols recommended that an organization should be publishing something they call news five days per week for at least six months, so the voters would know what they are voting for.  Those criteria might be modified, at least in some experimental jurisdictions, especially in news deserts, as something else is done to create local news organizations eligible to receive a portion of the experimental citizen-directed subsidies.  

The [[w:Institute for Nonprofit News|Institute for Nonprofit News]] and Local Independent Online News (LION) Publishers&lt;ref&gt;&lt;!-- Local Independent Online News (LION) Publishers--&gt;{{cite Q|Q104172660}}&lt;/ref&gt; help local news organizations get started and maintain themselves.  Organizations like them might help new local news initiatives in experimental jurisdictions as discussed in this article.

== Funding research in the value of local news ==

We would expect two sources of funding for research to quantify the value of local news 

* The [[w:World Bank|World Bank]] has already discussed the value of news. We would expect that organizations that fund community and local development projects would also want to fund experiments in anything that seemed likely to increase the return on their investments in such projects. 
* Folkenflik (2023) wrote, &quot;Some of the biggest names in American philanthropy have joined forces to spend at least $500 million over five years to revitalize the coverage of local news in places where it has waned.&quot; This group of philanthropic organizations includes the American Journalism Project, which says they &quot;measure the impact of our philanthropic investments and venture support by evaluating our efficacy in catalyzing grantees’ organizational growth, sustainability and impact.&quot;&lt;ref&gt;Website of the American Journalism Project accessed 29 December 2023 ([https://www.theajp.org/about/impact/# https://www.theajp.org/about/impact/#]).&lt;/ref&gt; 

If the claims made above for the value of news are fair, then appropriate experiments might be able to quantify who benefit from improving the news ecology, how much they benefit, which structures seem to work the best, and even the optimal level of funding.

== Summary == 

This article has summarized numerous claims regarding different ways in which information is a public good.  Many such claims can be tested in experiments crudely similar to those for which Banerjee, Duflo, and Kremer won the [[w:2019 Nobel Memorial Prize in Economic Sciences|2019 Nobel Memorial Prize in Economics]].  We suggest funding such projects as companions to Community and Local Development (CLD) projects.  If the research cited above is replicable, the returns on such investments could be huge, delivering benefits to the end of human civilization, similar to those claimed for newspapers published in the US in the early nineteenth century, which may have made major contributions to carrying the US to its current position of world leadership and to developing technologies that benefit the vast majority of humanity the world over today.

== Acknowledgements == 

Thanks especially to Bruce Preville who pushed for evidence supporting wide ranging claims of media influence in limiting progress against many societal ills.  Thanks also to Dave Black for suggesting experimenting with Native American jurisdictions in the US and to Joy Ellsworth for describing the substantial cultural challenges that such interventions might face.

== References == 

* &lt;!-- Craig Aaron (2021) &quot;Media reform per Freepress.net&quot;, Wikiversity--&gt;{{cite Q|Q118225069}}, accessed 2023-07-22.
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* &lt;!-- Antonela Miho (2020) Small Screen, Big Echo? Estimating the Political Persuasion of Local Television News Bias using Sinclair Broadcast Group as a Natural Experiment--&gt;{{cite Q|Q120895467}}
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* &lt;!-- US National Academies of Sciences, Engineering, and Medicine (2023) Nobel Prize Summit Fuels Initiatives to Combat Misinformation and Disinformation and Build Trust in Science--&gt;{{cite Q|Q124711722|author=National Academies}}
* &lt;!-- Ivan Natividad (2022) State funds Berkeley Journalism $25 million to strengthen California's local news coverage--&gt;{{cite Q|Q123537590}}
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* &lt;!--Kyle Pope (2023-04-17) &quot;Free Press, Functioning Democracy: You can’t have one without the other&quot;, Columbia Journalism Review--&gt;{{cite Q|Q124047437}}
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* James D. Wolfensohn (2002) Foreword to Islam et al. (2002, pp. v-vi).

== Notes ==
{{reflist}}
[[Category:Government]]
[[Category:News]] 
[[Category:Original research]]
[[Category:Research]]
[[Category:Political science]]
[[Category:Media]]
[[Category:Freedom and abundance]]
[[Category:Economics]]
[[Category:Political economy]]
[[Category:News]]
[[Category:Corruption]]
[[Category:Democracy]]</text>
      <sha1>pmuzih13o0red5lsproqpp2hi19kp8z</sha1>
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  </page>
  <page>
    <title>Linear and noble Boolean functions</title>
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    <revision>
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{| class=&quot;wikitable&quot; style=&quot;float: right; text-align: center; &quot;
! arity
! &lt;math&gt;n&lt;/math&gt;
! 1
! 2
! 3
! 4 
! 5
|-
! linear
| &lt;math&gt;2^{n+1}&lt;/math&gt;
| 4
| 8
| 16
| 32
| 64
|-
! noble
| &lt;math&gt;2^{2^{n-1}}&lt;/math&gt;
| 2
| 4
| 16
| 256 
| 65536
|}

Among the truth tables for a given arity, the linears and the [[Noble Boolean function|nobles]] are important subsets.

Each linear can be assigned a patron, which is noble. Each noble can be assigned a prefect, which is linear.

For arity 3 they form a bijection. For higher arities the nobles outnumber the linears &lt;small&gt;(i.e. the patrons of the linears are subset of the nobles)&lt;/small&gt;.


==overview==

{| class=&quot;collapsible-with-classes collapsible collapsed wide followed&quot;
!colspan=&quot;2&quot;| 2-ary
|-
|
{{multiple image
 | align = center | total_width = 500
 | image1 = 2T linears with quadrants.svg
 | image2 = 2Z linears with quadrants.svg
 | footer = linears &amp;nbsp; &lt;small&gt;(truth tables and Zhegalkin indices)&lt;/small&gt;
}}
| [[File:2-ary nobles in matrix.svg|thumb|center|245px|nobles]]
|}

{| class=&quot;collapsible-with-classes collapsible open wide followed&quot;
!colspan=&quot;2&quot;| 3-ary
|-
|
{{multiple image
 | align = center | total_width = 700
 | image1 = 3T linears with quadrants.svg
 | image2 = 3Z linears with quadrants.svg
 | footer = linears &amp;nbsp; &lt;small&gt;(truth tables and Zhegalkin indices)&lt;/small&gt;
}}
| [[File:3-ary nobles in matrix.svg|thumb|center|345px|nobles]]
|}

{{Collapsible START|4-ary|collapsed wide}}
{{Collapsible START|linears &amp;nbsp; &lt;small&gt;(truth tables)&lt;/small&gt;|collapsed wide light followed}}
[[File:4T linears with quadrants.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible START|linears &amp;nbsp; &lt;small&gt;(Zhegalkin indices)&lt;/small&gt;|collapsed wide light followed}}
[[File:4Z linears with quadrants.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible START|nobles|collapsed wide light}}
[[File:4-ary nobles in matrix.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible END}}

==3-ary==

{| class=&quot;collapsible-with-classes collapsible collapsed wide light gap-below&quot;
!colspan=&quot;3&quot; style=&quot;background-color: #f0f0f0; color: gray;&quot; | nobles in tesseract
|-
| [[File:3-ary nobles in tesseract.svg|500px]]
| [[File:2T principality; faction size 1; king index 0.svg|thumb|center|280px|dihedral symmetry]]
| [[File:2T principality; faction size 3; king index 2.svg|thumb|center|280px|mirror symmetry]]
|-
|}

{| class=&quot;collapsible-with-classes collapsible open wide followed&quot;
!colspan=&quot;2&quot; bgcolor=&quot;#ddd&quot; | quadrants
|-
|style=&quot;padding-right: 15px;&quot;| [[File:3-ary linear to noble, quadrant 0.svg|thumb|right|300px| {{colorbox|#e30000}} &amp;nbsp; 0 &amp;nbsp; &lt;small&gt;(even, evil)&lt;/small&gt;]]
|style=&quot;padding-left: 15px;&quot;| [[File:3-ary linear to noble, quadrant 3.svg|thumb|left|300px| {{colorbox|#00cc00}} &amp;nbsp; 3 &amp;nbsp; &lt;small&gt;(odd, odious)&lt;/small&gt;]]
|-
|style=&quot;padding-right: 15px;&quot;| [[File:3-ary linear to noble, quadrant 2.svg|thumb|right|300px| {{colorbox|#ffb000}} &amp;nbsp; 2 &amp;nbsp; &lt;small&gt;(even, odious)&lt;/small&gt;]]
|style=&quot;padding-left: 15px;&quot;| [[File:3-ary linear to noble, quadrant 1.svg|thumb|left|300px| {{colorbox|#0055ff}} &amp;nbsp; 1 &amp;nbsp; &lt;small&gt;(odd, evil)&lt;/small&gt;]]
|}

{| class=&quot;collapsible-with-classes collapsible collapsed wide followed&quot;
!colspan=&quot;2&quot; bgcolor=&quot;#ddd&quot; | halves
|-
|style=&quot;padding-right: 15px;&quot;| [[File:3-ary linear to noble, even.svg|thumb|right|300px|even&lt;br&gt;&lt;small&gt;Walsh functions&lt;/small&gt;]]
|style=&quot;padding-left: 15px;&quot;| [[File:3-ary linear to noble, odd.svg|thumb|left|300px|odd&lt;br&gt;&lt;small&gt;complements of Walsh functions&lt;/small&gt;]]
|}

{| class=&quot;collapsible-with-classes collapsible collapsed wide&quot;
!colspan=&quot;2&quot; bgcolor=&quot;#ddd&quot; | all
|-
| [[File:3-ary noble to linear.svg|300px|center]]
|}


==4-ary==

===linear to patron (noble)===

The patrons of the 32 linears are 32 nobles. Their indices are the 3-ary Boolean functions with consul 0.

{{Collapsible START|quadrants 0 and 3|collapsed wide followed}}
{| class=&quot;wikitable&quot; style=&quot;float: right; text-align: center;&quot;
|- style=&quot;font-size: 65%;&quot;
! king
| 0
| 3808
| 26752
|-
! king index
| 0
| 14
| 104
|}
{{colorbox|#e30000}} &amp;nbsp; The patrons of the 8 even Walsh functions &lt;small&gt;(quadrant 0)&lt;/small&gt; are the red entries in these 3 principalities.&lt;br&gt;
{{colorbox|#00cc00}} &amp;nbsp; The patrons of their complements &lt;small&gt;(quadrant 3)&lt;/small&gt; are the green entries.
{| class=&quot;collapsible-with-classes collapsible collapsed light wide followed&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| truth tables
|-
| [[File:3T principality; faction size 1; king index 0.svg|400px]]
| [[File:3T principality; faction size 6; king index 14.svg|400px]]
| [[File:3T principality; faction size 1; king index 104.svg|400px]]
|}
{| class=&quot;collapsible-with-classes collapsible collapsed light wide&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| Zhegalkin indices
|-
| [[File:3Z principality; faction size 1; king index 0.svg|400px]]
| [[File:3Z principality; faction size 6; king index 14.svg|400px]]
| [[File:3Z principality; faction size 1; king index 104.svg|400px]]
|}
{{Collapsible END}}

{{Collapsible START|quadrants 1 and 2|collapsed wide}}
{| class=&quot;wikitable&quot; style=&quot;float: right; text-align: center;&quot;
|- style=&quot;font-size: 65%;&quot;
! king
| 5760
| 10920
|-
! king index
| 22
| 42
|}
{{colorbox|#ffb000}} &amp;nbsp; The patrons of the 8 odious Walsh functions &lt;small&gt;(quadrant 2)&lt;/small&gt; are the yellow entries in these 2 principalities.&lt;br&gt;
{{colorbox|#0055ff}} &amp;nbsp; The patrons of their complements &lt;small&gt;(quadrant 1)&lt;/small&gt; are the blue entries.
{| class=&quot;collapsible-with-classes collapsible collapsed light wide followed&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| truth tables
|-
| [[File:3T principality; faction size 4; king index 22.svg|400px]]
| [[File:3T principality; faction size 4; king index 42.svg|400px]]
|}
{| class=&quot;collapsible-with-classes collapsible collapsed light wide&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| Zhegalkin indices
|-
| [[File:3Z principality; faction size 4; king index 22.svg|400px]]
| [[File:3Z principality; faction size 4; king index 42.svg|400px]]
|}
{{Collapsible END}}

===noble to prefect (linear)===

...


[[Category:Boolean functions; Zhegalkin stuff]]</text>
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  </page>
  <page>
    <title>User:Dc.samizdat/A symmetrical arrangement of 120 11-cells</title>
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      <text bytes="120208" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from the 60-point (hemi-icosahedral cell) for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemiicosahedron rotation.gif|ps=; &quot;The [[W:Hemi-cube|hemi-cube]] is an abstract polyhedron with a real presentation as a uniform star cube with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted 12-point (Jessen's icosahedron) vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure), the real hexad cell of the 11-point (11-cell). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted Jessen's icosahedron vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to form a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's larger concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

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[[File:Hemicube.svg|thumb|The [[W:Hemi-cube|hemi-cube]] is an abstract polyhedron with a real presentation as a uniform star cube with 5 faces (3 squares and 2 triangles), 12 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Christie|2024|loc=File:Pentahemiicosahedron rotation.gif|ps=; &quot;The [[W:Hemi-cube|hemi-cube]] is an abstract polyhedron with a real presentation as a uniform star cube with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.&quot;.}}]]

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from the 60-point (hemi-icosahedral cell) for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemiicosahedron.png|ps=; &quot;The '''pentahemiicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted 12-point (Jessen's icosahedron) vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure), the real hexad cell of the 11-point (11-cell). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted Jessen's icosahedron vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to form a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's larger concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

[[File:Hemicube.svg|thumb|The [[W:Hemi-cube|hemi-cube]] is an abstract polyhedron with a real presentation as a uniform star cube with 5 faces (3 squares and 2 triangles), 12 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Christie|2024|loc=File:Pentahemiicosahedron.png|ps=; &quot;The '''pentahemiicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.&quot;.}}]]

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from the 60-point (hemi-icosahedral cell) for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted 12-point (Jessen's icosahedron) vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure), the real hexad cell of the 11-point (11-cell). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted Jessen's icosahedron vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to form a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's larger concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

[[File:Hemicube.svg|thumb|The [[W:Hemi-cube|hemi-cube]] is an abstract polyhedron with a real presentation as a uniform star cube with 5 faces (3 squares and 2 triangles), 12 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.&quot;.}}]]

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Citation|title=120-cell|title-link=W:120-cell|journal=Wikipedia|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2010}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from the 60-point (hemi-icosahedral cell) for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted 12-point (Jessen's icosahedron) vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure), the real hexad cell of the 11-point (11-cell). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted Jessen's icosahedron vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to form a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's larger concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

[[File:Hemicube.svg|thumb|The [[W:Hemi-cube|hemi-cube]] is an abstract polyhedron with a real presentation as a uniform star cube with 5 faces (3 squares and 2 triangles), 12 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.&quot;.}}]]

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from the 60-point (hemi-icosahedral cell) for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted 12-point (Jessen's icosahedron) vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure), the real hexad cell of the 11-point (11-cell). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted Jessen's icosahedron vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to form a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's larger concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

[[File:Hemicube.svg|thumb|The [[W:Hemi-cube|hemi-cube]] is an abstract polyhedron with a real presentation as a uniform star cube with 5 faces (3 squares and 2 triangles), 12 edges, and 5 vertices, the contracted [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the [[W:11-cell|11-cell]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.&quot;.}}]]

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from the 60-point (hemi-icosahedral cell) for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two triangles lie opposed, inscribed in a hexagon central plane.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted 12-point (Jessen's icosahedron) vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure), the real hexad cell of the 11-point (11-cell). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted Jessen's icosahedron vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to form a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's larger concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two triangles lie opposed, inscribed in a hexagon central plane.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted 12-point (Jessen's icosahedron) vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure), the real hexad cell of the 11-point (11-cell). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted Jessen's icosahedron vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to form a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's larger concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Citation|title=24-cell|title-link=W:24-cell|journal=Wikipedia|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024}}
* {{Citation|title=600-cell|title-link=W:600-cell|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|journal=Wikipedia}}
* {{Citation|title=120-cell|title-link=W:120-cell|journal=Wikipedia|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2010}}
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      <text bytes="119599" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two triangles lie opposed, inscribed in a hexagon central plane.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted 12-point (Jessen's icosahedron) vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure), the real hexad cell of the 11-point (11-cell). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted Jessen's icosahedron vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to form a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's larger concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even ascribe to it a distinct dimensionality (such as 4, 3, or 5), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two triangles lie opposed, inscribed in a hexagon central plane.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted 12-point (Jessen's icosahedron) vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure), the real hexad cell of the 11-point (11-cell). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted Jessen's icosahedron vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to form a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's larger concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even ascribe to it a distinct dimensionality (such as 4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two triangles lie opposed, inscribed in a hexagon central plane.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted 12-point (Jessen's icosahedron) vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure), the real hexad cell of the 11-point (11-cell). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted Jessen's icosahedron vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to form a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's larger concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two triangles lie opposed, inscribed in a hexagon central plane.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted 12-point (Jessen's icosahedron) vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), an inverted Jessen's icosahedron vertex figure of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to form a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's larger concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two triangles lie opposed, inscribed in a hexagon central plane.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.gif|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's abstract concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
* {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two triangles lie opposed, inscribed in a hexagon central plane.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell, and 12 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges contribute one {{radic|5}} triangle face each to the cell's abstract concentric partner the 60-point (truncated icosahedron), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two triangles lie opposed, inscribed in a hexagon central plane.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two triangles lie opposed, inscribed in a hexagon central plane.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles lie opposite 4 hexagons; 2 triangles are not opposite each other. The quasi-regular 11-cell is a compound of 5 pentad cells and 6 of this hexad cell, a 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. It is an expanded 6-point (hemi-cuboctahedron) with 4 triangular faces and 4 hexagonal faces. On a {{radic|2}}-radius 3-sphere, 6 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges each contribute one of their {{radic|5}} triangle faces to an 11-cell.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Citation|title=600-cell|title-link=W:600-cell|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|journal=Wikipedia}}
* {{Citation|title=120-cell|title-link=W:120-cell|journal=Wikipedia|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2010}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two triangles lie opposed, inscribed in a hexagon central plane.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
* {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 5 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 5 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two opposed triangles are inscribed in two completely orthogonal hexagon central planes.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 7 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 7 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two opposed triangles are inscribed in two completely orthogonal hexagon central planes.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Hemicube.svg|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 7 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 7 vertices, and an expansion of the hemi-cube, 5 vertices into 7 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two opposed triangles are inscribed in two completely orthogonal hexagon central planes in 4-space, so they do intersect at one point, the center of the 3-sphere.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Citation|title=16-cell|title-link=W:16-cell|journal=Wikipedia|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024}}
* {{Citation|title=24-cell|title-link=W:24-cell|journal=Wikipedia|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024}}
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      <text bytes="119550" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 5-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract polyhedron with 5 faces (3 squares and 2 triangles), 15 edges, and 7 vertices. It is a contraction of the 11-point ([[W:11-cell|11-cell]]) 4-polytope, 11 vertices into 7 vertices, and an expansion of the hemi-cube, 5 vertices into 7 vertices. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. The two opposed triangles are inscribed in two completely orthogonal hexagon central planes in 4-space, so they do intersect at one point, the center of the 3-sphere.&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
* {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!elevenad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:10-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract heptad polyhedron with 8 faces (3 squares and 5 triangles), 15 edges, and 7 vertices. It is an expansion of the 5-point (hemi-cube). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two 7-point (heptad) cells meet at a triangular face to form the 11-point (11-cell) 4-polytope, which contains five 5-point (5-cells).&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract heptad polyhedron with 8 faces (3 squares and 5 triangles), 15 edges, and 7 vertices. It is an expansion of the 5-point (hemi-cube). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two 7-point (heptad) cells meet at a triangular face to form the 11-point (11-cell) 4-polytope, which contains five 5-point (5-cells).&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract heptad polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices. It is an expansion of the 5-point (hemi-cube). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two 7-point (pentahemicosahedron) cells meet at a triangular face to form the 11-point (11-cell) 4-polytope, which contains five 5-point (5-cells).&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Citation|title=16-cell|title-link=W:16-cell|journal=Wikipedia|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024}}
* {{Citation|title=24-cell|title-link=W:24-cell|journal=Wikipedia|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024}}
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      <text bytes="119428" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract heptad polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices. It is an expansion of the 5-point (hemi-cube). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two 7-point (pentahemicosahedron) cells merge their orthogonal 4-edge paths to form the 11-point (11-cell) 4-polytope, which contains five 5-point (5-cells).&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
* {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract heptad polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices. It is an expansion of the 5-point (hemi-cube). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells meet at their orthogonal 4-edge path to form the 11-point (11-cell) 4-polytope, which contains five 5-point (5-cells).&quot;.}}
|-
!120
!100
!120
|}
That's all that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''pentahemicosahedron''' or '''[[W:Hemi-cube|hemi-cube]]''' is an abstract heptad polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices. It is an expansion of the 5-point (hemi-cube). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells meet at their orthogonal 4-edge path to form the 11-point (11-cell) 4-polytope, which contains five 5-point (5-cells).&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to form the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells).&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

Each elevenad building block is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed). There are 120 elevenad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Citation|title=600-cell|title-link=W:600-cell|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|journal=Wikipedia}}
* {{Citation|title=120-cell|title-link=W:120-cell|journal=Wikipedia|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2010}}
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      <text bytes="120163" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to form the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells).&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The 7-point '''Pentahemicosahedron''' is the heptad building block, an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
* {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to form the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells).&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The 7-point '''Pentahemicosahedron''' is the heptad building block, an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to form the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells).&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point [[W:hemi-icosahedron|hemi-icosahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point [[W:hemi-icosahedron|hemi-icosahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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      <text bytes="119812" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 120 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

And here at last we find the formulae... ...relating &lt;math&gt;pentad orthorschemes^4&lt;/math&gt; to &lt;math&gt;hexad orthorschemes^4&lt;/math&gt;... ...for surely every object in 4-space has both in some characteristic relation... ...the &lt;math&gt;=&lt;/math&gt; sign in its conservation law by Noether's theorem...

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following, then push it all off into subsequent sections (including all the illustrations). then reduce it to a short summary to be put here to conclude this section. 

.... 

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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* {{Citation|title=24-cell|title-link=W:24-cell|journal=Wikipedia|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024}}
* {{Citation|title=600-cell|title-link=W:600-cell|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|journal=Wikipedia}}
* {{Citation|title=120-cell|title-link=W:120-cell|journal=Wikipedia|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2010}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following, then push it all off into subsequent sections (including all the illustrations). then reduce it to a short summary to be put here to conclude this section. 

=== ... ===
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
* {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following, then push it all off into subsequent sections (including all the illustrations). then reduce it to a short summary to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is used only when we build the great whale, the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following, then push it all off into subsequent sections (including all the illustrations). then reduce it to a short summary to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!240
|}
That's everything that's in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is used only when we build the great whale, the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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      <text bytes="119934" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!240
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|2}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is used only when we build the great whale, the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
* {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!240
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 240 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is used only when we build the great whale, the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is used only when we build the great whale, the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is used only when we build the great whale, the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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      <text bytes="120178" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''Pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is used only when we build the great whale, the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is used only when we build the great whale, the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is always present (and required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}}]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is only present (and always required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1970 | title=Twisted Honeycombs | place=Providence, Rhode Island | journal=Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics | publisher=American Mathematical Society | volume=4 }}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this illustration. The viewer must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is only present (and always required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this illustration. The viewer must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is only present (and always required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The viewer must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture of its contents, on the back of the box of building blocks:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is only present (and always required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The viewer must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It's all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is only present (and always required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The viewer must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is only present (and always required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The viewer must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block lies completely orthogonal to another pentad building block in the 120-cell. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is only present (and always required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|dc@samizdat.org}}
{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The viewer must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block in the 120-cell lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges that run through a central plane of the 3-sphere to get from one side to the other.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is only present (and always required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1970 | title=Twisted Honeycombs | place=Providence, Rhode Island | journal=Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics | publisher=American Mathematical Society | volume=4 }}
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The viewer must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block in the 120-cell lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is only present (and always required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{align|center|April 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The viewer must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block in the 120-cell lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is only present (and always required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a convex 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[W:120-cell|120-cell]], the largest regular convex 4-polytope, which also contains inscribed instances of all the regular convex 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells and 120 11-cells. The 11-cell is a quasi-regular 4-polytope, containing 11 pentad regular 4-polytope cells and 11 hexad semi-regular 3-polytope cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view. The viewer must imagine them.]]
The 4-dimensional regular convex 4-polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:simplex|simplex]] (called the [[W:5-cell|5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:orthoplex|orthoplex]] (called the [[W:16-cell|16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the 16-point ([[W:8-cell|8-cell 4-cubic]]), which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found as a building block in any of the other regular 4-polytopes, except in the largest and most complex one, the [[W:120-cell|120-cell]], the biggest thing you can build with this set of building blocks (the picture on the cover of the box, which is built from everything in the box). But we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== The 5-cell and the hemi-icosahedron in the 11-cell ==
The most apparent relationship between the hemi-icosahedron and the 5-cell is that they both have 10 triangular faces. When we see a facet congruence between a 3-polytope and a 4-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 5-simplex (pentad). It also has 6 vertices like the 3-orthoplex (hexad); thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are famously ''only'' related to each other indirectly by dimensional analogies, having no face congruences in 4-space. The 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals (pentad and hexad). Interesting!

The 5-cell and 11-cell of radius {{radic|2}} both have edge length {{radic|5}}, and their triangular faces are congruent. There are 11 disjoint 5-cells inscribed in the 11-cell, sharing 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. This is a bit hard to see, because the hemi-icosahedron is hard to see, but it will be easier to see once we have identified the hemi-icosahedron and the 11-cell's elements exactly where they actually reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each 5-cell has 10 edges from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share an edge. Each hemi-icosahedron's 15 edges come from 10 distinct 5-cells, and there is just one 5-cell with which it does not share an edge. It turns out that 5-cells and ''hemi-icosahedra'' meet edge-to-edge, but not face-to-face, even though 5-cells and ''11-cells'' do meet face-to-face!

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in different planes, completely orthogonal to each other on the 3-sphere. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness|2022|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes.]]
We shall see in the next section that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software{{Sfn|Moxness|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} using Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s which can render the polyhedra which are found in the interior of ''n''-dimensional polytopes. [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the quadrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his hull #8 with 60 vertices, lower right. It is the real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (hull #8) is the concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[w:600-cell|600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in these central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing irregular great hexagons (truncated great triangles) of alternating 120-cell edges and 5-cell edges. There are 10 great dodecagons and 60 5-cell edges in Moxness's 60-point (hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (hull #8) polyhedron altogether.

Moxness's 60-point (hull #8) is an irregular form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point (icosidodecahedron), which has the same relationship to Moxness's 60-point (hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|Notice that the 5-point (regular 5-cell) could be another abstraction of Moxness's 60-point (hull #8), 12-vertices-into-1, though it is not a ''contraction''. It will turn out that it is an ''expansion''.}} The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than quadrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. 

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (hull #8) is the real hemi-icosahedron, with 60 5-cell edges in it as chords lurking under its surface. It has enough of those 5-cell parts to make 20 faces, or 2 whole 5-cells, but it does not contain any of those objects. The 60 5-cell edges belong individually to neighboring 5-cells. 

The 600-point (120-cell) may be constructed from these 60-point (Moxness's rhombicosadodecahedra), joined edge-to-edge sharing a 5-cell edge. They won't fill the 3-sphere by themselves; holes will be left between them. In this construction the 120-cell is entirely filled by a honeycomb containing 60-point (rhombicosadodecahedra), 20-point (dodecahedra), and 10-point ([[w:Pentagonal_prism|pentagonal prism]]s). The 60-points meet each other at their triangle faces, meet the regular dodecahedra at their pentagon faces, and meet the pentagonal prisms at their rectangle faces. We might try to imagine such a honeycomb with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with the parts truncated from 60 neighboring regular dodecahedra that are pentagonal prisms). There are 120 of each kind of dodecahedron.

Before we move on from Moxness's 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice its other great circle polygons. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra, as those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, discovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|}}|name=all the polyhedra}} The 12 little pentagons are spanned by 5 sets of 6 orthogonal {{radic|5}} chords (5-cell edges). The 6 chords of each set are symmetrically arranged as 3 parallel pairs, not as far apart as they are long, in 3 orthogonal central planes; they form 3 orthogonal ''linked'' great rectangles. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, in the section below where we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[W:120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[W:120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[W:120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-points ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|ps=; This hexad of scholars from a New Orleans, Louisiana high school extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks of its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Symmetrical_5-set_Venn_diagram.svg|120px]]{{Sfn|Grünbaum|1976|loc=Rotationally symmetrical 5-set Venn diagram, 1975|ps=; 5-cell.}}
|[[File:Tetrahemihexahedron_rotation.gif|120px]]{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}
|[[File:Pentahemicosahedron.png|120px]]{{Sfn|Christie|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point '''Pentahemicosahedron''' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;.}}
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts you need to build the 120-cell (the picture on the cover of the box), and everything else that fits in the 120-cell: all the astonishing regular 4-polytopes.

Each regular 5-cell is a pentad building block. The pentads have {{radic|5}} edges. There are 120 pentad building blocks in the box.

Each pentad building block in the 120-cell lies completely orthogonal to another pentad building block. They are completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges.

Every vertex of the 24-cell, and therefore every vertex of the 600-cell and the 120-cell, has a 6-point (octahedral vertex figure). The hexad building block is that 6-point object embedded at the center of the 120-cell instead of at a vertex. The hexads have red triangle faces with {{radic|5}} edges, which are 5-cell faces. Their yellow square faces lie in central planes, because the hexad is an abstract 4-polytope. Its {{radic|6}} edges are the diagonals of the yellow square faces. There are 100 hexad building blocks in the box.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 60 pairings of a pentad with its completely orthogonal hexad, and there are two chiral ways of choosing the completely orthogonal pairs (left-handed or right-handed), so there are 120 11-cells.

The heptad building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|5}} edges and 9 {{radic|4}} edges), and 7 vertices. It is an expansion of the 5-point ([[W:hemi-cube|hemi-cube]]) and the 6-point ([[W:hemi-icosahedron|hemi-icosahedron]]). Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Two completely orthogonal 7-point (pentahemicosahedron) cells can be joined at their common Petrie polygon (a skew hexagon which contains their orthogonal 4-edge paths) to construct the 11-point ([[W:11-cell|11-cell]]) 4-polytope, which contains five 5-point (regular 5-cells). There are 120 heptad building blocks in the box.

=== Building with pentads + hexads ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell does indeed contain the 600 ''octahedral'' cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It is a 600-cell in two different ways, although the octahedra are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell.}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (5-cell), out of the 12-point (hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope we can make out of the 5-point (pentad 5-cell) building block beside the largest one, it just isn't a ''regular'' 4-polytope. It is quasi-regular which means it has two kinds of cells, analogous to the quasi-regular polyhedra: the 12-point (cuboctahedron) and the 30-point (icosidodecahedron), which have two kinds of faces. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of cells, it also means that it has cells of two different dimensionalities. The 11-cell has two kinds of cells, one of which is a 3-polytope building block (a hexad), as is usual for cells of 4-polytopes, and one of which is a 4-polytope building block (the 4-simplex regular 5-cell), which is almost unprecedented; we have seen that the regular 5-cell occurs nowhere else in the 4-polytopes except in the 120-cell. The 4-simplex cell is inscribed in the 11-cell in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, which is entirely different than the way 3-polytope cells occur as building blocks in 4-polytopes. Of course the 5-point (5-cell) contains five 4-point (3-polytope) cells, so the 11-cell is still quasi-regular with two varieties of 3-polytope cells (regular tetrahedra and truncated tetrahedra), but additionally it has two ''dimensionalities'' of ''n''-polytope cells: another clue that the 11-cell has something to do with dimensional analogy. The 11-cell has 5 completely disjoint pentad cells, and 6 volumetrically disjoint hexad cells. The pentad cells are 5 of 120 completely disjoint instances forming a compound of completely disjoint 5-point (4-polytopes), the 120-cell. Each 11-cell shares each of its hexad cells with 3 other 11-cells. The hexad cells are 6 of 200 never-disjoint instances forming a compound of non-disjoint 12-point (3-polytopes), the 120-cell, with each of its 600 vertices occurring in 4 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.}}

We said earlier that the 11-cell is a compound of 11 disjoint 5-cells which are inscribed in it, and that this might be hard to see, but now we are going to see numerically, at least, how it can have eleven cells. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, eleven. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (5-cell) building blocks into 120 5-point (5-cell) building block cells, and the 12 6-point (hexad) building blocks into 100 6-point (hexad) building block cells, and then magically adding one whole 5-point (5-cell) building block cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 6-point (hexad) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such 11-cell bundles, with each bundle containing 11 of each kind of cell. Very, very strange!

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like Euclid's postulates, and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as compounds of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles.{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated to generate its regular 4-polytope because it is the complete gene for it, containing all of its elements without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} 

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[W:5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the compound of 120 disjoint regular 5-cells, so it can be subdivided by &lt;math&gt;120\times 120 = 14400&lt;/math&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[W:16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its planes of symmetry at once intersecting at its center, so its symmetry is of order 384.  The 120-cell is the convex hull of the compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;.

=== One fibration of 11-cells ===
11-cells are not required (or permitted) in any of the objects built so far. The heptad building block is only present (and always required) when building the 600-point (120-cell).

There are 120 distinct 11-cells inscribed in the 120-cell, but unlike the 120 completely disjoint 5-cells,  none of the 11-cells are disjoint. There is only ''one'' 11-cells (plural) in the 120-cell. That is, there is only a single Hopf fibration of the great circles of the 11-cells. The 5-cell and the 11-cell are limit cases: the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. The 120-cell contains 0 disjoint 11-cells and 120 distinct 11-cells. The 11-cell is abstract only in the sense that no disjoint instances of it exist. The 11-cells exist, but the 11-cell singular exists only in the 120-cell, in the presence of 120 regular 5-cells and 119 other instances of itself.

== The 11-point 11-cell ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer|Segerman|2013|loc=Puzzling the 120-cell}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his [[W:120-cell#Visualization|section of the Wikipedia article on the 120-cell entitled ''Visualization'']].{{Sfn|Ruen|Goucher|Christie|Moxness|2010|loc=Goucher's ''§Visualization'' describes the decomposition of the 120-cell into rings two different ways; subsection ''§Intertwining rings'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), ''not the other way around''.

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the regular dodecahedron is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each great circle fiber of the 120-cell that occurs in that fiber bundle is conflated to one distinct point on the dodecahedron map.   

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. The obvious candidate is Moxness's 60-point (rhombicosadodecahedron), which is the hemi-icosahedral cell of the 11-cell. Here we return to our inspection of the rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

So let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. They do meet at a {{radic|5}} edge, however. We might have suspected at first that there is a pentagonal prism between two rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while two pentagonal prisms connected to the middle pentagon arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section, and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[W:120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen|2011|ps=; These encyclopedia articles contain only textbook knowledge. Ruen did not contribute any original research to them, of course. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are three of them, illustrating respectively: the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell, the 10-fold decagonal symmetry of the 10 120-points in the 120-cell, and the 11-fold polygonal symmetry of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three orthogonal projections of the 120-cell to a central plane, from the table of 14 30-grams
|-
!
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|120 pentads + 100 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 pentads + 6 hexads
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|20
|align=center|12
|align=center|11
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|20
|align=center|12
|align=center|1
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|- 
!style=&quot;white-space: nowrap;&quot;|In inscribed 4-polytopes
|align=center|225 24-cells
|align=center|10 600-cells
|align=center|120 11-cells
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|10
|align=center|60
|align=center|200
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|36°
|align=center|6°
|align=center|1.8°
|}

Here we summarize our findings on the 11-cell, including in advance some results to be obtained in the next section. 

...finish and organize the following ... section, pushing most or all of it into subsequent sections (including all the illustrations). then reduce everything to a short summary of findings to be put here to conclude this section. 

== ... ==
[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk|2018|loc=File:Polyhedron truncated 20 from yellow max.png|ps=; Truncated icosahedron.}}]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations and there is only one of it in the 600-point (120-cell). It also represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells. No part of the hologram is contained by any object smaller than the whole 120-cell. However, it has an analogous 60-point (32-face 3-polytope) that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 60-point (32-cell). Both 60-points have 12 pentad facets and 20 hexad facets, polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of completely orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere. 

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon....  

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur, and all 120 of them occur in the 120-cell, and nowhere else.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[W:Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|Rossiter|2022}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie|2024|loc=Bucky Fuller and the languages of geometry}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1948-1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with his mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the shad can open his six beaks all the way, until they become flat squares and he becomes a cuboctahedron, or he can shut them all tight like a turtle retracting into his octahedron shell. The six mouths always move in unison. This is [[W:Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[W:Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[W:Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do his ''really'' odd trick -- where he flips his 6 jaws 90 degrees in his 6 faces and shuts his 6 beaks on the opposite axis of their squares than the one he opened them on -- or whether he will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than his normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make his edge length exactly the same size as his radius, when he opens his mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing his mouths in spherical synchrony, his 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron, then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the Euclidean 3-polytope Jessen's, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. This is a not a normal orthogonal projection of that object, it is the object inverted.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|In Euclidean 4-space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; Hemi-cuboctahedron.}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces (the ''heptad'' arrives at the party). The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. The 12-point is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 200 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}}]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:5InscribedTruncatedtetrahedra.png|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of this 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. On a {{radic|2}}-radius 3-sphere, 5 of them with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell of the 11-cell. [20 of them with {{radic|5}} triangle edges and {{radic|6}}&lt;nowiki&gt; hexagon long edges are cells in the abstract quasi-regular 60-point (truncated icosahedral 4-polytope), a compound of 12 regular 5-cells.]&lt;/nowiki&gt;]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

...The 10 pairs of central edge-planes in the two 5-cells are completely orthogonal to each other....

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

........

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two completely orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[w:Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded (rhombicosadodecahedra), as if a monster 4-dimensional shadfish had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of octahederal&lt;math&gt;^3&lt;/math&gt; (16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[W:Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh|Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[W:24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. Fuller did include the tetrahedron in the contraction cycle after the octahedron, so perhaps he saw deeper than the [[W:Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and there may be a cycle (in every dimension) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider whether in such a cycle the shad would make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point.

== The 12-point Legendre  binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, it expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre) 3-polytope is embedded in Euclidean 4-space to remarkable effect. In 3-space [[W:Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction. In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them to each other. 

=== The ''n''-simplexes ===
....

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} great hexagons. These are the same 200 dodecagon central planes which also contain 6 120-cell edges and 6 regular 5-cell edges, forming two opposing irregular great hexagons, [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|as illustrated above]].]]
The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== The ''n''-orthoplexes ===
....
...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[W:600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[W:Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane.

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[W:16-cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[W:16-cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=all the polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[W:16-cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== The ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== The ''n''-equilaterals ===
A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[W:24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[W:24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== The ''n''-pentagonals ===
....{{Efn|1=Square roots of non-integers are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio &lt;math&gt;\tfrac{1}{\phi} = \phi^{-1}&lt;/math&gt;
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽&lt;sup&gt;2&lt;/sup&gt; = &lt;math&gt;\tfrac{1}{\phi^2} = \phi^{-2}&lt;/math&gt; ≈ 0.382&lt;br&gt;
For example:
{{indent|7}}𝝓 = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618
|name=fractional square roots|group=}}


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== The ''n''-leviathon ===
...concise [[W:120-cell#Cords|''Chords'']] diagram and abbreviated text of the [[W:120-cell#Relationships among interior polytopes|''Relationships among interior polytopes'']] section from the 120-cell article....my fan of major chords circle diagram, with a legend that is the leftmost and rightmost columns of the major chords table....the major chords are the ''n''-edges, their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge....

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre) polytope is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of 600 of its instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 120 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 12-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction,{{Sfn|Stott|...}} before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]]. Between Alicia Boole Stott and the identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and the identification of conservation laws with symmetry groups, we owe the deepest mathematics of geometry and physics originally to two pioneering women who were contemporaries and colleagues of the greatest men of the academy in their time, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just a singleton convex 4-polytope, not just a honeycomb,{{Sfn|Coxeter|1970}} and not just an [[W:abstract polytope|abstract 4-polytope]]. The 11-cell has a concrete realization as its identified element sets, which are subsets of the 120-cell's element sets just as all the 4-polytopes' element sets are. 

The 120 11-cells' realization as 600 Legendre 12-point 4-polyhedra captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=apology}}

== Regular convex 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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  </page>
  <page>
    <title>BIM-224 Research Infrastructures 24</title>
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      <contributor>
        <username>Loz.ross</username>
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      <text bytes="3780" xml:space="preserve">''Materials and Tasks for the module &quot;BIM-224, SoSe 2024, Blümel/Rossenova&quot; for students at Hochschule Hannover. The materials are prepared with several colleagues from the [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab at TIB] Hannover.''

=== Session 1: Data harvesting interfaces / data collection ===
Slides are available here: https://docs.google.com/presentation/d/1P0JECD0X7ceCtUOQqc3az3mrDSQrmHX-i-INe6p3ZzA/edit?usp=sharing

==== &lt;u&gt;Student homework task pages&lt;/u&gt; ====
* Student Name / link to wiki..
* Rama / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/RamaUnterlagen
* Davud / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/SeitevonDavud
* Enes / https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/EnesUnterlagen
* Lena | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/Lenskatala
* Aurora | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/lazydocs
* Burak | https://de.wikiversity.org/wiki/Kurs:OpenKnowledge24/BuraksUnterseite

==== &lt;u&gt;Group task 1&lt;/u&gt; ====

===== Platform list =====

* [https://ww2.bgbm.org/herbarium/default.cfm Herbarium Berolinense] (Herbarium des Botanischen Gartens und des Botanischen Museums Berlin)
* [https://codingdavinci.de/ Coding Davinci]
* Typographia Sinica used Dataset for Metadata is [[doi:10.22000/756|Doi]]
* Digitale Historische Bibliothek Erfurt / Gotha
* DFG Viewer
* [https://corpusvitrearum.de/cvma-digital/bildarchiv.html CORPUS VITREARUM Bildarchiv] 
* [https://radar4culture.radar-service.eu/ radar4cultur] 

===== Type of API list =====

* [https://sketchfab.com/developers Sketchfab API] for [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]

==== &lt;u&gt;Group task 2&lt;/u&gt; ====

* Name / dataset link
* Rama | https://creating-new-dimensions.org/Restaging-Fashion/
* Davud I https://creating-new-dimensions.org/Schlangenkoepfe-und-koerper/
* Lena | [https://creating-new-dimensions.org/Schriftprobensammlung/ Schriftensammlung des BGBM]
* Enes | [https://creating-new-dimensions.org/der-mensch-und-seine-dinge/ Der Mensch und seine Dinge]
* Aurora | [https://creating-new-dimensions.org/Die-Sammlung-wissenschaftlicher-Instrumente-und-Lehrmittel-der-ETH-Zuerich/ Sammlung Wissenschaftliche Instrumente und Lehrmittel]
* Burak | [https://creating-new-dimensions.org/houdon-buesten-in-3d/ Houdon-Büsten in 3D] / [https://codingdavinci.de/node/2020 TransformingAntiquity]

=== Session 2: Data cleaning, reconciliation and enrichment ===
Slides are available here: https://docs.google.com/presentation/d/1IgKsZ4awJslXmE7Im7czSjAm5aUhBAk8wLMzbPr5KUg/edit?usp=sharing

==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* ...
* Lena | [[:de:Datei:Lena_Schriftproben_BGBM.pdf|Schriftprobensammlung BGBM]]

=== Session 3: Data in Wikidata ===
Slides are available here: https://docs.google.com/presentation/d/13LdJnA_y673wod2uMZMfY2FWID0YcbpfNkMbpEuMgVY/edit?usp=sharing
==== &lt;u&gt;Group task 1:&lt;/u&gt; ====
* [Enes Albayrak/ https://de.wikiversity.org/wiki/Datei:EnesALbayrak.jpg] ...
*Davud Kilic I Aspidites melanocephalus (siehe Bild)[[File:Triple Aimé Bonpland.jpg|thumb|Triple Aimé Bonpland[[File:Aspidites melanocephalus.png|thumb|Triple Davud Kilic I Aspidites melanocephalus]]]]
*Lena | Aimé Bonpland (siehe Bild)


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* ...

=== Session 4: Data Upload and querying ===
Slides are available here: https://docs.google.com/presentation/d/1-zgB_ndBQlUQrWQt5-1kci458blMaVaKDy1tqEM5Up8/edit?usp=sharing


==== &lt;u&gt;Homework presentations:&lt;/u&gt; ====
* Student name / presentation link (google slides, other slide platform, or wiki pages with screenshots)
* ...

=== Session 5: Data upload and querying (cont.) / Data visualisation and presentation ===


=== FINAL SUBMISSION ===</text>
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  <page>
    <title>How might the world be different if the PLO had followed Gandhi?</title>
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      <text bytes="68581" xml:space="preserve">{{Research project}} 

:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].''  

::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' 

== Abstract == 

This article evaluates how the world might be different if the Palestine Liberation Organization (PLO), founded in 1964, had sought a redress of grievances through nonviolence rather than violence. This analysis rests on a summary of research comparing the relative effectiveness of violence and nonviolence and the role of the media in conflict. It concludes with four suggestions for ending the cycle of violence and building a better future for all: (1) Demand equal protection of the laws. (2) Limit &quot;state secrets privilege&quot; to make it harder for governments to deny equal protection and lie about it with impunity. (3) Support training in nonviolence for all. (4) Citizen-directed subsidies for local news nonprofits to make it harder for major media to encourage their audiences to support counterproductive actions.

== Introduction == 

How might the world be different if the [[w:Palestine Liberation Organization|Palestine Liberation Organization]] (PLO), founded in 1964, had been committed to nonviolence, following [[w:Mahatma Gandhi|Gandhi]], [[w:Martin Luther King Jr.|King]], and [[w:Abdul Ghaffar Khan|Badshah Khan]] rather than [[w:George Washington|George Washington]] and [[w:Fidel Castro|Fidel Castro]]?

Nothing can be said about this for certain, except that the world would be different. However, careful study of history suggests that the world would most likely be better for virtually all Jews and Palestinians.&lt;ref&gt;This might be regarded as '[[w:counterfactual history| counterfactual history]]&quot;, which, especially in analyses like the present, invites people to consider the implications of alternative approaches to problems in light of research on human behavior and political economy.&lt;/ref&gt; 

This should ''NOT'' be construed as a criticism of [[w:Yassir Arafat|Yassir Arafat]] nor of anyone who supported the PLO nor any other organization that has adopted violent tactics such as [[w:Hamas|Hamas]] since 2023-10-07: They were following the example of [[w:George Washington|George Washington]]. How could they go wrong? 

Our answer to this apparent contradiction, discussed briefly below, is that few of the violent revolutions since 1776 have had the success attributed to the American Revolution, because the subsequent success of the US was achieved ''in spite of'', rather than because of, the violence of the American Revolution. The traditional narrative of the American Revolution has been written to please people who control most of the money for the media -- to the detriment of everyone else. Over 50 percent of adult white males could vote before the revolution, and the violence of the revolution did not change that, as discussed below. 

Similarly, the nonviolence of the [[w:First Intifada|First Intifada]] led to the election of [[w:Yitzhak Rabin|Yitzhak Rabin]] as Prime Minister of Israel on a platform of negotiating with Palestinians. That led to the [[w:Oslo Accords|Oslo Accords]] and the current [[w:State of Palestine|State of Palestine]]. We claim that if the Palestinians had maintained nonviolent discipline, the two-state solution promised at Oslo would likely have worked to benefit all.

Few supporters of Israel have any substantive understanding of the extent of the mistreatment of Palestinians by the Israeli military and settlers. The nonviolence of the First Intifada convinced enough Israeli voters that they could live in peace with Palestinians that Yitzhak Rabin won an election in 1992 to become Prime Minister of Israel on a platform of negotiating with Palestinians. Most nonviolent campaigns have produced similar results, as discussed below. 

Tragically, subsequent violence by both sides has created obstacles to honest consideration by each of how their opposition perceives them. Palestinians during the First Intifada and since have seen throwing rocks as relatively nonviolent. That is clearly not how most supporters of Israel have perceived that. In 2022 the Israeli ambassador to United Nations [[w:Gilad Erdan|Gilad Erdan]] complained that the world has been silent in the face of Palestinian “terror attacks with rocks” against Israelis, as he held up a rock the size of a brick. He noted that a rock like that could kill someone in a car speeding along a highway.&lt;ref&gt;Erdan's complaint was reported seriously by Lazeroff (2022) in the ''Jerusalem Post'', but was ridiculed by Willliams (2022) in the ''Washington Report on Middle Eastern Affairs''. Pressman (2017) said that throwing rocks should be considered &quot;unarmed violence&quot;.&lt;/ref&gt; 

This suggests that the nonviolence of the First Intifada, discussed below, might have been more effective if Palestinians had not thrown rocks: The shift in Israeli public opinion that got Yitzhak Rabin elected as Prime Minister would likely have been greater, and the international pressure on Israel would also likely have been greater. 

A vigorous commitment to nonviolence has worked in the past, even within the conflict between Jews and Palestinians. It seems to offer the only realistic prospect for breaking the cycle of violence and building a better future for both Palestinians and Jews.  

{{Blockquote|text=
''Oh, would some Power the gift give us''

''To see ourselves as others see us!''

''It would from many a blunder free us''
|multiline=yes
|author = [[w:Robert Burns|Robert Burns (1786)]]
|title = [[w:To a Louse|''To a Louse'']]}}

== Research comparing violence and nonviolence == 

Twenty-first century research can help us estimate the probability distribution of alternative outcomes in violent and nonviolent conflict. Most relevant in this regard is the inventory of all the major violent and nonviolent governmental change efforts of the twentieth century compiled by Erica Chenoweth and Maria Stephan (2011). They identified over 200 violent revolutions and over 100 nonviolent campaigns, each of which attracted over 1,000 people at some point. 53 percent of the nonviolent campaigns were successful while only 25 percent of the violent revolutions were. 

[[File:Democratization 1 year after vs. 1 year before twentieth century revolutions.svg|thumb|upright=2|Figure 1. Democratization 1 year after (vertical scale) vs. 1 year before (horizontal scale) the end of twentieth century revolutions]]

Probably more important than the official success rate is the impact on democracy: Chenoweth and Stephan (2011) found that on average, nonviolent campaigns ''improved'' the level of democracy, while violent revolutions had no statistically significant impact on democracy. This was true whether the campaigns won or lost. The gains for democracy tended to be greater among the nonviolent campaigns that won than among those that lost. However, even the nonviolent campaigns that lost on average pushed their governments to be more democratic, to share power more broadly; see Figure 1.

Similarly, Chenoweth and Schock (2015) noted that the presence of a &quot;[[w:radical flank effect|radical flank]]&quot;, contemporary violence pursuing similar aims, tended to ''reduce'' the probability of success. See also Chenoweth (2016).

== The nonviolence of the First Intifada == 

The NAVCO 1.1 dataset&lt;ref&gt;Chenoweth (2019a).&lt;/ref&gt; compiled by Chenoweth and Stephan includes five campaigns in Palestine or involving Palestinians: 

# &quot;'''[[w:1936–1939 Arab revolt in Palestine|Palestinian Arab Revolt]]'''&quot; in Palestine against &quot;Pro-Jewish British policies&quot; 1936-1939 coded as violent with limited success but with no change in Polity IV scores. 
# &quot;'''[[w:Mandatory Palestine#Beginning of Zionist Insurgency|Jewish resistance]]'''&quot; in &quot;Palestinian Territories&quot; against &quot;British occupation&quot; 1945-1948 coded as a violent success with no change in Polity IV scores.&lt;ref&gt;There are minor differences between how this is coded in NAVCO 1.1 and the description found in Wikipedia on 2024-03-31. For example, the section on &quot;[[w:Mandatory Palestine#Beginning of Zionist insurgency|Beginning of Zionist insurgency]]&quot; in the Wikipedia article on &quot;[[w:Mandatory Palestine|Mandatory Palestine]]&quot; mentions the assassination of &quot;Lord Moyne in Cairo&quot; 1944-11-06, while Chenoweth and Stephan (2011) coded this campaign as starting in 1945, not 1944. This discrepancy might be explained, as Chenoweth and Stephan only included cases where they &quot;were certain that more than 1,000 people were actively participating in the struggle, based on various reports.&quot; Chenoweth and Stephan may not have been able to document &quot;more than 1,000 people&quot; in that struggle prior to 1945. See also Chenoweth (2019b).&lt;/ref&gt; Chenoweth and Stephan (2011, p. 304) reported that only &quot;three successful violent insurgencies were succeeded by democratic regimes: the National Liberation Army’s 1948 victory in Costa Rica, the Jewish resistance in British-occupied Palestine, and the 1971 Bengali self-determination campaign against Pakistan. However, these instances represent only three cases out of fifty-five successful insurgencies in the twentieth century. They are as rare as authoritarian regimes that succeed victorious nonviolent campaigns. This variation points to a potentially fruitful avenue of future research&quot;, such as experiments suggested below in the section on, &quot;Implications for the future&quot;. 
# &quot;'''[[w:Black September|Palestinian activists]]'''&quot; in Jordan violently contesting &quot;Jordanian rule&quot; in 1970 coded as a failure with a modest decline from (-9) to (-10) in Polity IV scores, shifting Jordan to the most authoritarian point on the Polity IV scale.&lt;ref&gt;This doubtless refers to &quot;[[w:Black September|Black September]]&quot;, which Wikipedia reports as having run from 1970-09-06 to 1971-07-23, while NAVCO 1.1 codes both the beginning and end as 1970. This difference seems negligible for present purposes.&lt;/ref&gt; 
# '''The [[w:First Intifada|[First] Intifada]]''' in Palestine against &quot;Israeli occupation&quot; 1987-1990, coded as a partial success from nonviolence but with no change in Polity IV scores.&lt;ref&gt;The end date for the &quot;Intifada&quot; in NAVCO 1.1 is different from the description in Chenoweth and Stephan (2011), whose chapter 5 is titled, &quot;The First Palestinian Intifada, 1987-1992&quot;. The end date in the corresponding Wikipedia article was 1993-09-13 (when checked 2024-03-31), different from both the end dates in NAVCO 1.1 and Chenoweth and Stephan. However, it seems that these differences can be safely ignored for present purposes. The coding of this campaign as &quot;nonviolent&quot; and a &quot;partial success&quot; is consistent with its contributions to the Oslo Accords and its subsequent failure to achieve the two-state solution promised by Oslo. The appearance of Palestinian violence late in that campaign is consistent with the discussion of the impact of a &quot;radical flank&quot; by Chenoweth and Schock (2015) and Chenoweth (2016).&lt;/ref&gt;
# '''The longer violent &quot;Palestinian Liberation&quot; campaign''' (1973- ) against &quot;Israeli occupation&quot; beginning in 1973 and still ongoing in 2006, coded as a failure with no change in Polity IV scores. 

:''The nonviolence of the First Intifada did more to move Israeli public opinion to believe that they could live in peace and harmony with Palestinians than anything else Palestinians have done since the 1917 [[w:Balfour Declaration|Balfour Declaration]]'', at least according to the literature that we've found credible. 

When the First Intifada began, Yitzhak Rabin was Israel's Minister of Defense. He could see that the nonviolence could not be suppressed with massive counter violence for two reasons: 
# Excessive violence against nonviolent demonstrators generated bad press that was actually moving Israeli&lt;ref&gt;Peri (2012) described how Israeli public opinion towards Palestinians softened as Palestinian terrorist attacks receded into history and hardened in response to violence targeting Jews. On p. 23, he said, “In the 1990s -- during the peace process, which made it appear that the era of warfare was at an end and that Israel was becoming a postwar society -- the professional autonomy of the media grew, and journalists adopted a more critical stance. However, the failure of the peace talks in the summer of 2000 and the outbreak of the second Intifada with its suicide attacks aimed at the heart of the civilian population led to a serious retreat ... . State agencies and the public even more so again exerted pressure for media reorientation, demanding that the media restrain its criticism and circle the wagons.&quot;&lt;/ref&gt; and international opinion.&lt;ref&gt;King (2007).&lt;/ref&gt;
# Rabin knew that ''he could not count on soldiers to follow orders'' if they perceived their orders as out of proportion to the provocations.&lt;ref&gt;Peri (1993) reported that, &quot;The Palestinian uprising in the Israeli-occupied territories that began in December 1987 poses challenges of an unprecedented nature and difficulty for Israeli society. One of those challenges comes in the form of a conscientious objection to perform military service. ... At the same time, however, some one hundred officers and noncommissioned soldiers have been tried and jailed for refusing to perform military service in the West Bank and Gaza Strip. In addition to them, several thousands are in a gray area of refusal. These latter are not put on trial, and therefore no report about them goes to the higher military authorities or the public.&quot; Similarly, Peri (1996, p. 355) said that as the Intifada continued, Rabin &quot;had begun, in conversations with those close to him, to speak of a dimension that he would not dare to expand on publicly: that the war against the Intifada was damaging the [w:Israeli Defense Forces|IDF]'s fighting spirit, hurting army morale, and undermining the status of the status of the IDF as a people's army.&quot; Also, the Wikipedia article on &quot;[[w:Refusal to serve in the Israel Defense Forces|Refusal to serve in the Israel Defense Forces]]&quot; lists several organizations that have appeared consisting of Israelis who are refusing to serve in occupied territories, e.g., in Lebanon in the late 1970s and in the West Bank and Gaza in the 1980s and 1990s.&lt;/ref&gt; 

Early in the Intifada, he had told his soldiers to shoot to wound, in the legs and feet. As the nonviolence and negative press continued, he issued clubs and ordered soldiers to beat people, breaking bones.&lt;ref&gt;Munayyer (2011).&lt;/ref&gt; Before the Intifada, Rabin had not wanted to talk with Palestinians, saying, &quot;There was no point&quot;, because they always had to check with King Hussein of Jordan or President Mubarak of Egypt or President Assad of Syria.&lt;ref&gt;Peri (1996, p. 353).&lt;/ref&gt; That changed with the Intifada, because the Palestinians &quot;proved that for the first time in their history, they had decided to take charge of their fate.&quot;&lt;ref&gt;Peri (1996, p. 356).&lt;/ref&gt;

When the nonviolence continued, Rabin ran for Prime Minister on a platform of negotiating with Palestinians. He became Prime Minister in 1992 ''and was reportedly pleased when his staff told him he would not have to negotiate with leaders of the nonviolence.''&lt;ref&gt;Shlaim (2014, p. 533): &quot;Rabin’s conversion to the idea of a deal with the PLO was clinched by four evaluations ... .First ... a settlement with Syria was attainable but only at the cost of complete Israeli withdrawal from the Golan Heights. Second ... the local Palestinian leadership had finally been neutralized. Third ... Arafat’s dire situation, and possible imminent collapse, made him the most convenient interlocutor ... . Fourth ... the impressive progress achieved through the Oslo channel. Other reports that reached Rabin during this period pointed to an alarming growth in the popular following of Hamas and Islamic Jihad in the occupied territories [which] stressed to him the urgency of finding a political solution&quot;. See also King (2007, ch. 12).&lt;/ref&gt; Israeli leaders were desperate. They sent in ''[[w:agent provocateur|agents provocateurs]]'', who were exposed and neutralized until Israel expelled 481 leaders of the nonviolence and arrested between 57,000 and 120,000 others. Finally Israel got the violence they needed to justify overwhelming counter violence. That Palestinian violence also brought the ultra-Zionist right wing parties back to power in Israel.&lt;ref&gt;Mary Elizabeth King (2007, 2009) and Wikipedia, [[w:First Intifada|First Intifada]], accessed 2023-03-35.&lt;/ref&gt;

==Violence and nonviolence in the American Revolution== 

The &quot;[[w:Age of Revolution|Age of Revolution]]&quot; (1765-1849, including the French Revolution and the Latin American revolutions of the nineteenth century), plus the [[w:Russian Revolution|Russian]],&lt;ref&gt;There were actually two Russian Revolutions in 1917, the &quot;[[w:February Revolution|February]]&quot; and &quot;[[w:October Revolution|October]]&quot; Revolutions. The first was mostly nonviolent, resulting from massive popular displeasure with the management of the Russian political economy by the Tsar during World War I. The second was a violent reactions of the failures of the government that replace the Tsar, leading to the Russian Civil War. This sequence of events is crudely comparable to the First Intifada, in that the potential success of each was cut short by violence, with tragic consequences. We argue that a better popular understanding of nonviolence would likely have produced better results for both with much less loss of life.&lt;/ref&gt; [[w:Chinese Communist Revolution|Chinese]],&lt;ref&gt;The Wikipedia article on &quot;[[w:Chinese Revolution|Chinese Revolution]]&quot; lists several revolutions, all violent, none enhancing democracy. The longest and most consequential was the [[w:Chinese Communist Revolution|Chinese Communist Revolution]] (1927-1949). That's the one that would come first to many people's minds. However, it was not the only one.&lt;/ref&gt; and [[w:Cuban revolutions|Cuban]] revolutions, as well as the violent post-World War II anti-colonial struggles of Africa and Asia&lt;ref&gt;See the section on &quot;[[w: Decolonization#After 1945|After 1945]]&quot; in the Wikipedia article on &quot;[[w:Decolonization|Decolonization]]&quot;, accessed 2024-03-25.&lt;/ref&gt; all, from at least some perspectives, replaced one brutal repressive system with another. Many, perhaps all, have supporters who claim that common folk benefitted from that violence. For example, Napoleon introduced the [[w:Napoleonic Code|Napoleonic Code]], which has had a major influence on the civil code in many countries around the world, including the US state of Louisiana and most of Latin America and Eurasia.&lt;ref&gt;Wikipedia, &quot;[[w:List of national legal systems|List of national legal systems]], accessed 2024-04-01.&lt;/ref&gt; However, claims that any of this violence made substantive advances for freedom and democracy, liberty and justice are at best controversial. 

The standard narrative of the American Revolution seems to suggest that the American Revolution was different from all those other attempts to emulate it: The US, according to this narrative, got freedom and democracy, liberty and justice for all from the violence of the American Revolution. 

The reality is more nuanced: The US got independence from Great Britain. However, claims that the US got more than that from the violence are controversial and largely contradicted by the available evidence. Gaughan (2022) notes that at the time of the Revolution, Great Britain was a constitutional monarchy, which was extremely unusual during a global era of autocracy. &quot;In the British Isles, only 15 to 20 percent of English men could vote. In contrast, ... [t]he rate of enfranchisement varied from colony to colony. ... [A]s many as 80 percent of men could vote in some colonies but only 50 to 60 percent in other colonies. ... During the Revolutionary era, most states expanded suffrage to at least some degree.&quot; This occurred via ''nonviolent'' democratic deliberation as the existing legislatures of the 13 rebelling colonies either wrote state constitutions or revised their colonial charters to delete reference to Great Britain. The research of Chenoweth and Stephan (2011) suggests that the advances for democracy would likely have been greater if the rebellious colonists had refused to use violence. After the [[w:Boston Tea Party|Boston Tea Party]] in 1773, Parliament ended local self government in Massachusetts. Then crowds of worried farmers largely prevented judges and others appointed the King from doing anything unless they promised to ignore the recent acts of Parliament and abide by the colonial charter, under which those judges and other officials were answerable to the locals, ''not'' to London.&lt;ref&gt;Raphael (2002).&lt;/ref&gt; If that kind of nonviolent response had been the dominant feature of the American Revolution instead of violence, the experience of Gandhi, King, Badshah Khan, and others described by Chenoweth and Stephan (2011), Chenoweth and Schock (2015), and Chenoweth (2016) suggests that the impact of the American Revolution on world history would likely have been greater.&lt;ref&gt;See also [[The Great American Paradox]] and Graves (2005).&lt;/ref&gt;

== Other research on nonviolence ==

Previous nonviolent campaigns have often succeeded by inventing new methods of protest as needs and opportunities were identifed. After an approach obtains some level of success, people with power often develop countermeasures that reduce the effectiveness of a known technique going forward.  

Gene Sharp documented 198 nonviolent tactics.&lt;ref&gt;Sharp (1973).&lt;/ref&gt; [[w:Nonviolence International|Nonviolence International]] maintains a growing database expanding Sharp's list of 198. By 2021, Nonviolence International had documented 148 more.&lt;ref&gt;Beer (2021, p. 7; 15/116 in pdf).&lt;/ref&gt;

== Role of the media in war == 

It has been said that the first casualty of war is truth.&lt;ref&gt;Knightly (2004) attributes this to US Senator Hiram Johnson in 1917. However, the consensus in multiple articles in Wikiquote seems to attribute it to [[Wikiquote:Philip Snowden, 1st Viscount Snowden|Philip Snowden]] in his introduction to Morel (1916, p. vii).&lt;/ref&gt; We suggest, however, that 

:''Truth typically dies long before anyone is killed in armed conflict.''

In most and perhaps virtually all conflicts, each party seems to believe that their actions are justified by atrocities committed by their designated enemies. 

:''Collateral damage that &quot;they&quot; commit proves to us that &quot;they&quot; are subhuman or at best criminally misled and must be resisted by any means necessary. Meanwhile, collateral damage that we commit is unfortunate but necessary -- from our perspective.''

However, to supporters of our opposition, the collateral damage that we have committed proves to them that we are subhuman or at best criminally misled and must be resisted by any means necessary. This asymmetry of perceptions is amplified by the media each party consumes: Every media organization sells changes in audience behaviors to the people who give them money. A media organization with no audience has nothing to sell. If they have an audience but displease their funders, they may not continue to have the money needed to produce the content required to retain an audience.&lt;ref name=ConfBHow2know&gt;Wikiversity, &quot;[[Confirmation bias and conflict]]&quot; and &quot;[[How can we know?]]&quot;, accessed 2024.03.26.&lt;/ref&gt;  

Media organizations everywhere mislead their audiences. They can do this easily, because everyone prefers information and sources consistent with preconceptions, a phenomenon called &quot;[[w:confirmation bias|confirmation bias]].&quot; Media organizations everywhere exploit confirmation bias to please those who control most of the money for the media.&lt;ref name = ConfB&gt;Wikiversity, &quot;[[Confirmation bias and conflict]]&quot;.&lt;/ref&gt; Wolfsfeld et al. (2002) noted that, &quot;The news media remain important agents for demonizing enemies and transforming political and military leaders into heroes. ... News is fundamentally ethnocentric, especially news about enemies ... because they threaten us.&quot; And violence attracts more attention than nonviolence. &quot;Battles are considered newsworthy, but ideas for preventing battles are not.&quot; Also, peace negotiations are usually conducted in extreme secrecy, because excessive publicity of negotiating positions could reduce the chances of success. Once an agreement is reached, the negotiators representing different parties to the conflict must then sell the agreement to their supporters. Prime Minister Rabin and King Hussein collaborated in supporting each other in selling the [[w:Israel–Jordan peace treaty|1994 Israel–Jordan peace treaty]] to their respective peoples. 

A review of relevant literature identified multiple drivers of increases in political polarization in recent decades:&lt;ref&gt;For more on this see Wikiversity, &quot;[[Information is a public good: Designing experiments to improve government]]&quot;, accessed 2024-04-01.&lt;/ref&gt;
* Increased concentration of ownership of the media, exemplified by the creation of ''[[w:Israel Hayom|Israel Hayom]]'' by [[w:Sheldon Adelson|Sheldon Adelson]] in Israel and its impact on Israeli politics (discussed below),&lt;ref&gt;Grossman et al. (2022).&lt;/ref&gt; as well as [[w:Vincent Bolloré|Bolloré]] in France,&lt;ref&gt;Cagé (2022).&lt;/ref&gt; [[w:Rupert Murdoch|Murdoch]] in Australia, the UK and the US,&lt;ref&gt;Murdoch's business focus is exemplified in the settlement Fox accepted in ''[[w:Dominion Voting Systems v. Fox News Network|Dominion v. Fox]]'': Fox admitted that they had initially reported honestly that Biden had won the 2020 US Presidential elections. However, after finding that they were losing audience to election deniers, they switched to reporting false claims about Dominion. Fox agreed to pay Dominion $787.5 million, provided they did not have to apologize for having lied to their audience. If lying about the 2020 election increased their audience and revenue by 6% in 2021, they made money, even after paying $787.5 million to Dominion. Fraud can be good business. Media executives could be fired if they lose money trying to protect democracy.&lt;/ref&gt; and Sinclair in the US.&lt;ref&gt;Ellison (2024), Kaviani et al. (2022), Miho (2020). This trend has been extended historically to include exclusive access offered by [[w:Western Union|Western Union]], founded in 1851, to the [[w:Associated Press|Associated Press]] (AP) as long as AP reporters did not criticize major corporations and monopolies and the contribution of those biases to the rise of the [[w:Robber baron|Robber baron]]s in the US in the late nineteenth century, according to McChesney (2004, pp. 35-36): &quot;[E]conomic historians regard the growth of Western Union as a major factor in the dominance of big business in American life. ... Western Union used its monopoly power to collaborate in the development of the [[w:Associated Press| Associated Press]] [AP, founded 1846], a monopoly news service run in cooperative fashion with the largest newspaper publishers. ... With exclusive access to the wires -- Western Union refused to let potential competitors use its wires -- AP became the only wire news service in the nation. ... Needless to say, [AP] invariably presented a voice that took the side of business interests.&quot;&lt;/ref&gt; Increased concentration of ownership of the media in apparently free societies make them look more like the unfree press in authoritarian regimes like Saudi Arabia, where journalists are more often incarcerated and killed.&lt;ref&gt;[[w:Assassination of Jamal Khashoggi|On 2018-10-02 Saudi government agents killed Jamal Khashoggi]], a Saudi journalist working for the Washington Post, because they did not like his reporting. Similarly, Khouri (2024) reported that, &quot;For the past six months, Israel has put a lot of effort into covering up its genocidal crimes in Gaza. One of the most brutal ways it does this is by routinely threatening, targeting and assassinating Palestinian journalists. The US-based Committee to Protect Journalists (CPJ) has reported that at least 90 Palestinian journalists have been killed since October 7 alongside two Israelis and three Lebanese. This is the highest death toll of journalists in any modern conflict that CPJ has monitored.&quot;&lt;/ref&gt; 
* The abolition of the [[w:Fairness doctrine|Fairness doctrine]] by the US [[w:Federal Communications Commission|Federal Communications Commission]] (FCC) in 1987 and the subsequent adaptation of the major media in the US to the belief systems of their increasingly distinct audiences. ([[w:Market segmentation|Segmentation]], a standard business practice, produces political polarization threatening democracy when applied to media markets, especially when control is too concentrated.) 
* The rise of the &quot;click economy&quot;, with Internet companies, especially social media like Facebook, making money from clicks, with algorithms that exploit confirmation bias in ways that increase political polarization, herding people into echo chambers in which people become increasingly convinced of the rectitude of their own positions and unknowingly increasingly ignorant of and insensitive -- and too often hostile -- to the constructed realties of people with whom they disagree.&lt;ref&gt;Carter (2021) and Wikiversity, &quot;[[Information is a public good: Designing experiments to improve government]]&quot;, accessed 2024-04-01.&lt;/ref&gt; 
* The decline in the quantity and diversity of local news, including the growth of news deserts and ghost newspapers.&lt;ref&gt;Abernathy (2020).&lt;/ref&gt; The decline in trusted local source(s) makes it easier for people to be misled by increasingly homogenized and biased corporate media and click bait,&lt;ref&gt;Darr et al. (2018, 2021), Zuboff (2018), Frenkel and Kang (2021), Vaidhyanathan (2018).&lt;/ref&gt; even threatening the national security of the US and its allies according to retired Lt. General [[w:H. R. McMaster|McMaster]],&lt;ref&gt;McMaster (2020).&lt;/ref&gt; former President Trump's second National Security Advisor.&lt;ref&gt;For a discussion of changes like these in Germany, see Floßer (2024). He discussed the [[w:Alternative for Germany|Alternative for Germany]] (Alternatif für Deutschland, AfD), a right-wing populist, political party in Germany, that insists that [[w:German collective guilt|Germany should not feel shame or guilt]] from what it did when Hitler was their leader. Floßer said the AfD generally got more votes ''in places with no local newspaper.''&lt;/ref&gt;&lt;ref&gt;For more on the decline of local news, its impact, and what to do about it, see the section on [[International Conflict Observatory#The current legal environment for Internet and other media companies amplifies political polarization and conflict|The current legal environment for Internet and other media companies amplifies political polarization and conflict]] in the Wikiversity article on [[International Conflict Observatory]].&lt;/ref&gt; 

The abolition of the US Fairness Doctrine clearly had no legal implications outside the US. However, Grossman et al. (2022) documented an apparently similar &quot;sea change in the right’s dominance of national politics&quot; in Israel that primarily benefited Benjamin Netanyahu and his Likud party following the 2007 launch of ''[[w:Israel Hayom|Israel Hayom]]'' by billionaire casino magnate [[w:Sheldon Adelson|Sheldon Adelson]]. It was distributed for free, reportedly to skirt Israel's campaign finance laws. &quot;[I]t soon became the most widely read newspaper nationally&quot;, which Grossman et al. attribute to this newspaper. Their &quot;findings highlight the immense impact the ultrarich can exert in shaping politics through media ownership.&quot;&lt;ref&gt;See also  Lalwani (2022). Similar concerns about the impact on French politics of the media empire of billionaire Vincent Bollaré are expressed in Cagé (2022).&lt;/ref&gt; This shift was described by US Senator [[w:Bernie Sanders|Bernie Sanders]], himself Jewish, saying, &quot;the Israel of today is not the Israel of … 20 to 30 years ago ... . It is a right-wing country, increasingly becoming a religious fundamentalist country where you have some of these guys in office believe that God told them they have a right to control the entire area.&quot;&lt;ref&gt;Hawkinson (2024).&lt;/ref&gt; Sander's comment is supported by Adelson's support for the claim that, &quot;the Palestinians are an invented people&quot;,&lt;ref&gt;Blumenfeld (2011).&lt;/ref&gt; a claim similar to the Zionist trope that Palestine was &quot;[[w:A land without a people for a people without a land|a land without a people for a people without a land]]&quot;.&lt;ref&gt;The extent to which this phrase helped drive the Zionist movement of the late nineteenth and twentieth centuries is controversial. See the Wikipedia article on &quot;[[w:A land without a people for a people without a land|a land without a people for a people without a land]]&quot;, accessed 2024-04-03.&lt;/ref&gt; 

''Israel Hayom'' is often called &quot;Bibiton&quot;, which is a portmanteau of &quot;Bibi&quot;, a nickname for Benjamin Netanyahu, and &quot;iton&quot;, the Hebrew word for newspaper. Adelson was accused&lt;ref&gt;Fulbright and Surkes (2017).&lt;/ref&gt; but not charged in the on-going corruption trial against Netanyahu&lt;ref&gt;Wikipedia, &quot;[[w:Trial of Benjamin Netanyahu|Trial of Benjamin Netanyahu]]&quot;. Netanyahu and his supporters have been working to undermine the judiciary to protect themselves from the issues raised in this trial, as documented in the Wikipedia article on &quot;[[w:2023 Israeli judicial reform|2023 Israeli judicial reform]]&quot;, accessed 2024-03-27. However, the issues raised in this legal battle are not as obviously related to questions of the role of the media in conflict, the topic of the present discussion.&lt;/ref&gt; and died before he was scheduled to testify in that trial.&lt;ref&gt;Alterman (2021).&lt;/ref&gt;

A 2022 survey in Israel found that ''Israel Hayom'' had the largest weekday readership exposure of any newspapers in Israel at 31%. The second and third most popular newspapers were ''[[w:Yedioth Ahronoth|Yedioth Ahronoth]]'' with 23.9% and ''[[w:Haaretz|Haaretz]]'' with 4.7% readership exposure.&lt;ref&gt;Readership figures are from a Hebrew-language document cited in the Wikipedia article on &quot;[[w:Newspapers in Israel|Newspapers in Israel]]&quot;, accessed 2024-04-03.&lt;/ref&gt; ''Yedioth Ahronoth'' tends to support former Israeli minister [[w:Tzipi Livni|Tzipi Livni]], who has been a leading Israeli politician advocating restraining expansion of Jewish settlements on the West Bank and negotiating with Palestinians; anything like this has been vigorously opposed by the editorial policies of ''Israel Hayom''. However, her support for Palestinians had limits, as indicated by a warrant for her arrest that was reportedly issued by a British court under universal jurisdiction, following an application by lawyers acting for Palestinian victims of the [[w:Gaza War (2008–2009)|2008-2009 Gaza war]]&lt;ref&gt;Also called &quot;Operation Cast Lead&quot;.&lt;/ref&gt; for her role in that operation as Israeli Foreign Minister. (The warrant was reportedly dropped with apologies from British political leaders after her visit to the UK was canceled.)&lt;ref&gt;Black and Cobain (2009).&lt;/ref&gt;

Political polarization doubtless exists between and within different groups supporting Palestinians and Israel driven by differences in the media they consume. Those differences in perception help perpetuate the conflict to the detriment of all.

== Implications for the future == 

What might be done to break the cycle of violence that has plagued Palestine and Israel since the 1917 Balfour declaration? 

Several ideas come to mind: 

=== Equal protection of the laws === 

Can enough Palestinians publicly and effectively renounce violence that it actually moves Israeli and international public opinion enough to force Israel to provide something like equal protection of the laws? Israel was moved in that direction by the nonviolence of the First Intifada. It could happen again. 

It would be great if Palestinians and / or Israelis and their supporters took the lead, but others do not have to wait for that. 

Rep. Marjorie Taylor Greene (R-GA) has introduced several motions in the US House of Representatives to try to censure Rep. Ilhan Omar for having said, &quot;From the river to the sea, Palestine will be free.&quot; Greene claimed that phrase was a Hamas slogan and &quot;calls for genocide of all Jews.&quot;&lt;ref&gt;Bahney (2023).&lt;/ref&gt; This has created problems for several Jewish members of the Congressional Progressive Caucus, who oppose Omar's use of that phrase but support her against censure on free speech grounds.&lt;ref&gt;Grayer (2024).&lt;/ref&gt; If Omar and others could ask instead for &quot;equal protect of the laws&quot;, it would be harder for people (like Greene) to oppose and easier for Jews and others to support.&lt;ref&gt;For more background on the use and interpretation of that and similar phrases by different groups, see the Wikipedia article on &quot;[[w:From the river to the sea|From the river to the sea]]&quot;, accessed 2024-04-09.&lt;/ref&gt;

Less well known is the portion of [[w:From the river to the sea|the 1977 election manifesto of the right-wing Israeli Likud party]] that, &quot;Between the sea and the Jordan there will only be Israeli sovereignty.&quot;&lt;ref&gt;Likud Party (1977).&lt;/ref&gt; Whether or not Ilhan Omar was intending to call for genocide of Jews, the current Israeli government, whose minister leads [[w:Likud|Likud]], has been accused of ''actual'' genocide in a complaint to the [[w:International Court of Justice|International Court of Justice]].&lt;ref&gt;[[w:South Africa v. Israel (Genocide Convention)|''South Africa v. Israel'' (2023)]].&lt;/ref&gt;

Perhaps it's better to avoid phrases like that entirely and instead demand something like &quot;equal protection of the laws.&quot; That principle was promised by the [[w:Fourteenth Amendment to the United States Constitution|Fourteenth Amendment to the US Constitution]], passed in 1868 and is still far from being implemented.&lt;ref&gt;The phrase &quot;equal protection of the laws&quot; are the last 5 words of [[w:Fourteenth Amendment to the United States Constitution#Section 1: Citizenship and civil rights|Section 1 of that amendment]]. That amendment has reportedly been the most frequently litigated part of the Constitution, and Section 1 has reportedly been the most frequently litigated part of the amendment. Just such a case active as this is being written in ''Johnson v. Parson'' (2024).&lt;/ref&gt; 

Let's broaden this question: What percent of the enemies of any country are primarily a result of routine denial of equal protection of the laws by people with power? The documentation summarized here suggests that the Palestinian violence since the Oslo Accords has largely been due to mistreatment, extended incarcerations without charges or trial, destruction of property, taking property at gun point, and even murder by Israeli security forces and settlers, unreported or underreported in the major media consumed by supporters of Israel, as noted by [[w:Ofer Cassif|Ofer Cassif]], a Jewish member of the Knesset. He said that in 2011 Israeli Prime Minister Netanyahu released [[w:Yahya Sinwar|Yahya Sinwar]], the violent leader of Hamas, in a prisoner exchange, and did ''not'' release [[w:Marwan Barghouti|Marwan Barghouti]], who has been called a Palestinian Mandela. Evidently, ''nonviolence'' is a much bigger threat than violence to the 1977 Likud promise, &quot;Between the sea and the Jordan there will only be Israeli sovereignty.&quot;&lt;ref&gt;Cassif et al. (2024).&lt;/ref&gt; Similarly, there is substantial documentation of routine denial of equal protection of foreigners by the US&lt;ref&gt;The [[w:1998 United States embassy bombings|1998 United States embassy bombings]] &quot;are widely believed to have been revenge for U.S. involvement in the extradition and alleged torture of four members of Egyptian Islamic Jihad (EIJ) who had been arrested in Albania in the two months prior to the attacks for a series of murders in Egypt&quot;, according to the Wikipedia article on those bombings, accessed 2024-04-22. If that's accurate, then the embassy bombings and the subsequent [[w:September 11 attacks|September 11 attacks]]  might not have occurred without US complicity in torture. And the September 11 attacks might not have occurred if the US had treated the embassy bombings as law enforcement issues. Beyond that, there is substantial documentation of US interference in foreign countries, e.g., overthrowing democratically elected governments to favor US international business interests. See, e.g., the Wikipedia article on [[w:Foreign interventions by the United States|Foreign interventions by the United States]] and references cited therein.&lt;/ref&gt; and France,&lt;ref&gt;See, e.g., the Wikipedia articles on [[w:Françafrique|Françafrique]] and [[w:François-Xavier Verschave|François-Xavier Verschave]] (accessed 2024-04-22), whose books titled &quot;''Françafrique'' (1999) and ''Noir silence'' (2020) have become standard works for anyone interested in the Rwandan genocide specifically, and generally the dissimulated policies followed by the French Republic in former colonies.&quot;&lt;/ref&gt; to name only two countries for which substantial documentation is available. For example, the [[w:September 11 attacks|suicide mass murders of September 11, 2001]] were major crimes but not acts of war by a major power capable of seriously threatening the United States. How might history since 2001 have been different if the US had treated the September 11 attacks as law enforcement issues and not an excuse to go to war? 

There is substantial documentation suggesting that the government of Afghanistan would likely have complied with a standard extradition request. Evidence is usually provided with extradition request,&lt;ref&gt;See the Wikipedia article on &quot;[[w:Extradition|extradition]]&quot;, accessed 2024-04-22, and references cited therein.&lt;/ref&gt; and the US refused to provide evidence when asking Afghanistan to extradite bin Laden. The documentation of the willingness of Afghanistan to consider such a request includes evidence that in 1998, June or July, the government of Afghanistan had agreed to extradite bin Laden to Saudi Arabia to stand trial for treason. If that extradition commitment had actually been carried out, bin Laden would have almost certainly been convicted and executed in typical Saudi justice. 

However, the transfer was set for September, because the Afghans needed time to separate bin Laden from his armed entourage. Then [[w:1998 United States embassy bombings|on August 7, the US embassies in Kenya and Tanzania were bombed]], and muslim clerics all over the world condemned those attacks as unjustified violence, reflecting ill on Islam. For 13 days. 

Then [[w:Operation Infinite Reach|the US bombed a pharmaceutical plant in Sudan and al-Qaeda training camps in Afghanistan]]. Muslim public opinion turned 180 degrees, as many concluded that bin Laden was correct: The US ''is'' an evil empire. The extradition was cancelled: Evidently, Afghan officials decided that if the US was not going to respect international law, maybe the world needs bin Laden. That thought was not restricted to Afghanistan. Donations to bin Laden ''increased'' substantially after the US retaliation for the embassy bombings. Employees of the Saudi embassy and consulates in the US began to support bin Laden's preparations for what became the suicide mass murders of September 11, 2001.

Afghan policy reportedly flipped again after the [[w:USS Cole bombing|bombing of the USS ''Cole'', 2000-10-12]]: Afghan officials agreed to inform the Clinton administration of bin Laden's whereabouts, so he could be killed by a US air strike. 

However, that plan was late in the Clinton administration. Implementation was passed to the incoming George W. Bush administration, which failed to act on this agreement before September 11, 2001, almost a year after the attack on the ''Cole''. Afghan officials seemed puzzled by the delay, reportedly offering in jest to provide fuel for the US aircraft to be used to kill bin Laden. 

In sum, before September 11, 2001, US government officials knew that the Afghan government had offered to help identify the whereabouts of bin Laden so he could be killed. They also knew that the government of Saudi Arabia had been involved in the preparations for the September 11 attacks. It was clear that the attacks were organized by a non-governmental entity that was not capable of threatening the internal security of the US. But rather than invading Saudi Arabia, and rather than treating Afghan officials with respect, Bush administration officials manipulated the international media to justify invading Afghanistan and Iraq. 

Senator [[w:Bob Graham|Bob Graham]] later said that during the hearings organized by the US House and Senate [[w:Joint Inquiry into Intelligence Community Activities before and after the Terrorist Attacks of September 11, 2001|Joint Inquiry into Intelligence Community Activities before and after the Terrorist Attacks of September 11, 2001]], the FBI went &quot;beyond just covering up ... into ... aggressive deception.&quot;&lt;ref&gt;Hulse (2015).&lt;/ref&gt; And then-President George W. Bush successfully convinced the committee to redact the results of that investigation from their December 2002 report. Those redactions became known as &quot;[[w:The 28 pages|The 28 pages]]&quot;. They were declassified over 13 years later by then-President Obama. During those 13 years, the US and allies were stampeded into invading Iraq by claims in news reports produced by reporters and editors, who should have known at the time were questionable and likely fraudulent.&lt;ref&gt;Similar media policies have driven other wars. These excesses were particularly egregious during and after [[w:World War I|World War I]], as documented by Hochschild (2022). The [[w:Espionage Act of 1917|Espionage Act of 1917]] gave the [[w:United States Postmaster General|Postmaster General]] the authority to declare as &quot;unmailable&quot; any publication that he decided did not adequately support the war effort. This effectively terminated many publications, because there was no other way to distribute publications nationally at that time (p. 61). The lack of broad discourse in the media amplified war hysteria, under which many people were persecuted, beaten, robbed, incarcerated, and even killed with impunity for peaceably assembling and petitioning for better wages and working conditions, or even for just speaking German. That continued after the war to help major capitalists suppress labor organizers.&lt;/ref&gt; 

What changes in the US political economy might reduce the ability of any administration to stampede the public, and the US Congress in particular, into supporting similar military actions without first exhausting non-military options?  

The US Congress could, for example, allow anyone anywhere to file suit in any US federal court for violations of ''equal protection'' by the US or Israel anyplace on earth.

=== Limit state secrets privilege === 

Equal protection of the laws cannot be guaranteed without limiting the ability of governments to deny equal protection in secret. In the US, this would require modifying the law governing &quot;[[w:State secrets privilege|state secrets privilege]]&quot;. 

Under past and current US secrecy practices, US government officials have provoked foreign entities to do things that were then denounced as unprovoked to justify counterproductive uses of military force. Current US government secrecy practices encourage such dangerous behaviors, according to documentation in Connelly (2023).&lt;ref&gt;See also the summary in Connelly, Samuelson, and Graves (2023).&lt;/ref&gt; The US responses to the 1998 embassy bombings and the September 11 attacks are examples, as discussed above. The 1997 report of the [[w:Moynihan Commission on Government Secrecy|Moynihan Commission on Government Secrecy]] reached conclusions similar to those of Connelly. Lying to Congress may officially be illegal, but exposing such lies has been punished severely while people who deceived Congress were rewarded, as documented with the revelations of [[w:Edward Snowden|Edward Snowden]]. Worse, [[w:Richard Barlow (intelligence analyst)|Richard Barlow]]'s career was reportedly destroyed merely for telling his managers they should not lie to Congress.  He did ''not'' expose officially classified government secrets, as had [[w:Whistleblowing|whistleblowers]] like Snowden: He came to public attention after filing suit for wrongful termination. 

[[w:Daniel Hale|Daniel Hale]] was sentenced to 45 months in prison for leaking a document that showed that during a five-month operation in Afghanistan, &quot;nearly 90% of the people killed during one five-month period ... were not the intended targets.&quot; Rep. Ilhan Omar asked President Biden to pardon Hale, because ''none of the documents released actually threatened US national security but instead &quot;shone a vital light on the legal and moral problems of the drone program and informed the public debate on an issue that has for too many years remained in the shadows.&quot;''&lt;ref&gt;Johnson (2021).&lt;/ref&gt; Similar things can be said about [[w:Daniel Ellsberg|Daniel Ellsberg]], [[w:Reality Winner|Reality Winner]], [[w:Chelsea Manning|Chelsea Manning]], [[w:Sibel Edmonds|Sibel Edmonds]], [[w:Jeffrey Alexander Sterling|Jeffrey Sterling]], and [[w:Thomas A. Drake|Thomas Drake]]. The attempted prosecution of [[w:Julian Assange|Julian Assange]] is similar but different, because Assange was never a US government employee and was rarely if ever in the US.&lt;ref&gt;See also Graves (2014, 2021).&lt;/ref&gt; 

The need for that much secrecy is further challenged by the research by Tetlock and Gardner (2015), who reported that their &quot;superforecasting&quot; teams &quot;performed about 30 percent better than the average for intelligence community analysts who could read intercepts and other secret data&quot; in forecasting problems assigned by the US Central Intelligence Agency (CIA).&lt;ref&gt;Tetlock and Gardner (2015, p. 95).&lt;/ref&gt; If important intelligence questions can be answered without government secrecy, that reinforces the conclusions of Connelly (2023) and others, that 

:* ''excessive government secrecy threatens more than advances US national security.''&lt;ref&gt;Connelly (2023); see also the summary in Connelly, Samuelson and Graves (2023).&lt;/ref&gt; 

We need government secrecy for some things, e.g., design and production technologies for sophisticated weapon systems and military plans for future operations. However, as the Moynihan Commission and Connelly reported, that's a tiny fraction of the information currently held as government secrets. Both claimed that the US would be safer and more prosperous if it limited government secrets to the things where government secrecy is really important. 

How can we do this? 

We suggest here that the US should authorize any federal judge to subpoena federal government documents that are classified as government secrets and declassify any, subject to appellate review, if the judge concludes that the public interest would be better served by declassification than by continued secrecy. The judge could issue a ruling with a rationale that would remain classified for a certain number of years. The ruling could go for or against the government. In a case like Ellsberg's, the judge could dismiss a criminal case with prejudice, which would preclude the government from filing charges again in front of a different judge. The judge could also award attorneys fees and damages.

=== Support training in nonviolence === 

As noted above, the nonviolence of the First Intifada led to the election of Yitzhak Rabin as Prime Minister of Israel on a platform of negotiating with Palestinians. This, in turn, led to the Oslo Accords and the current State of Palestine. As the time since the [[w:2023 Hamas-led attack on Israel|Hamas attacks on Israel of 2023-10-07]] has increased, more people in Israel,&lt;ref&gt;Wilkinson and Yam (2024).&lt;/ref&gt; the US and internationally have become increasingly concerned that the ferocity of Israeli attacks on Gaza (and the West Bank) seems far beyond the national security needs of Israel and out of proportion to the provocation.&lt;ref&gt;Mackenzie and Al-Mughrabi (2024).&lt;/ref&gt; Similarly, people are concerned that Israeli military attacks in Lebanon&lt;ref&gt;The Wikipedia article on [[w:2024 in Lebanon|2024 in Lebanon]] listed 8 attacks by Israel that killed at least 15 people in the first three months of 2024, when checked 2024-04-02.&lt;/ref&gt; and Syria&lt;ref&gt;Israel conducted lethal air strikes against alleged Iranian targets in Syria -- Aleppo March 29 and Damascus April 1, per Reuters (2024) and Bowen (2024); see also [[w:2024 Israeli bombing of the Iranian embassy in Damascus|2024 Israeli bombing of the Iranian embassy in Damascus]], accessed 2024-04-03.&lt;/ref&gt; threaten unnecessary expansion of the war.&lt;ref&gt;Peri (2020) claimed that Netanyahu's government had shifted more to the right in recent years, increasing the gap between military and political leaders. It could be helpful to have an update on how this split has impacted the current Israel-Hamas war and vice versa.&lt;/ref&gt; A new nonviolent movement led by Palestinians could strengthen people in Israel and elsewhere who are opposed to the current war.  

Palestinians and others could ask the US and Israel to support training in nonviolent civil disobedience for anyone interested, including designated &quot;terrorists&quot;. A demand like this should be difficult for people to oppose and could make it easier for people in many places to effectively work towards equal protection of the laws and other things. 

However, it is currently a criminal violation of the USA [[w:Patriot Act|Patriot Act]] to provide such training to anyone designated as a &quot;terrorist&quot; by the US State Department, per the Supreme Court decision in ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'' (2010). That law should be changed to ''support'' rather than ''criminalize'' such training.

How might the world be different if the US had vigorously supported rather than criminalized training Hamas and other designated &quot;terrorists&quot; in nonviolence? In particular, what are the chances that the [[w:2023 Hamas-led attack on Israel|Hamas-led attack on Israel of 2023-10-07]] would have happened if Palestinians could have seen progress in response to nonviolent protests against many of the outrages documented by the [[w:Palestinian Centre for Human Rights|Palestinian Centre for Human Rights]] and others? Currently few supporters of Israel seem to have any substantive awareness of the many outrages against Palestinians committed by people they support, because the media they find credible rarely if ever provides a balanced account of such outrages. If the US government had more openly supported such training, nonviolent protests facilitated by that training would likely have gotten better coverage in Israeli and international news. That in turn would likely have made it harder for the Israeli government to continue depriving Palestinians of equal protection of the laws so egregiously.

=== Citizen-directed subsidies for local news nonprofits === 

The Wikiversity article on &quot;[[Information is a public good: Designing experiments to improve government]]&quot; suggests randomized controlled trials to evaluate the long-term impact of citizen-directed subsidies for local news nonprofits distributed via local elections, as recommended by McChesney and Nichols (2021, 2022). They suggest distributing 0.15% of Gross Domestic Product (GDP) to local news non-profits via local elections with a limit on the maximum that any one local news organization could receive. Evidence summarized in that Wikiversity article suggests that the increases in political polarization in many countries in recent years may be due in part to a loss of local news, as advertising money has increasingly shifted to Internet companies that hire few if any journalists. 

To apply this to Israel and Palestine, we first note that their nominal GDPs in 2021 were $482 and $18 billion US$, respectively, totaling $500 billion.&lt;ref&gt;UN (2023).&lt;/ref&gt; 0.15% of that is $750 million. That may sound like a lot of money, but it's only 3.2% of the 2022 military budget of Israel of $23.4 billion.&lt;ref&gt;SIPRI (2024).&lt;/ref&gt; If such subsidies make a substantive contribution to reducing the political polarization that has driven this conflict for more than a century, it should help build broadly shared peace and prosperity for the long term for Israelis and Palestinians. If that happens, it would likely be the best investment in national security that Israel has made at least since 1948. This conjecture rests on the claims made above about news deserts, the &quot;click economy&quot;, and ''Israel Hayom''. If this works as predicted, it would also provide a shining example of new social technology that could help diffuse many other seemingly intractable conflicts around the globe.

=== Conclusions === 

We are not advocating an answer but a methodology that leading organizations have used successfully to build effective marketing campaigns worth billions of dollars: (i) Start by enunciating an objective like building broadly shared peace and prosperity while ending a cycle of violence. (ii) Brainstorm alternative approaches. (iii) Evaluate and refine them in focus groups. (iv) Test market the ones that seem most promising, and (v) go with what seems to work, (vi) while continuing to monitor results and adjusting accordingly. 

You, dear reader, may have other ideas that might be more effective than any of the things discussed herein to break the cycle of violence and build broadly shared peace and prosperity for the long term.

== Bibliography ==

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* &lt;!-- Darr, Hitt, and Dunaway (2018) &quot;Newspaper Closures Polarize Voting Behavior&quot;, J. of Communication--&gt;{{cite Q|Q61311391}} 
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* &lt;!-- Sarah Ellison (2024-02-16) &quot;Sinclair's recipe for TV news: Crime, homelessness, illegal drugs&quot;, Washington Post--&gt;{{cite Q|Q125271605}}
* Maxim Floßer (2024-03-06) &quot;Keine Lokalzeitung -- mehr AfD&quot;, Kontext--&gt;{{cite Q|Q125287792}}
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* &lt;!-- Alexander Fulbright and Sue Surkes (2017-06-26) &quot;Adelson testifies for second time in Netanyahu corruption probe&quot;, Times of Israel --&gt;{{cite Q|Q125303432}}
* &lt;!-- Anthony J. Gaughan (2022-12-30) Voting in Colonial and Revolutionary America--&gt;{{cite Q|Q125260096}}
* &lt;!-- Graves (2005) Violence, Nonviolence, and the American Revolution--&gt;{{cite Q|Q58635938}} 
* &lt;!-- Graves (2014) Restrict secrecy more than data collection--&gt;{{cite Q|Q106512569}}
* &lt;!-- Graves (2021) Congressional Gold Medals for Assange, Hale, Barlow, Winner, Manning, Edmonds, Sterling, Drake, Snowden, Ellsberg--&gt;{{cite Q|Q125570226}}
* &lt;!-- Annie Grayer (2024-03-29) &quot;Progressive Jewish lawmakers caught in the middle of bitter Democratic divide over Israel&quot;, CNN--&gt;{{cite Q|Q125247599}}
* &lt;!-- Guy Grossman, Yotam Margalit, and Tamar Mitts (2022-10) &quot;How the Ultrarich Use Media Ownership as a Political Investment&quot;, The Journal of Politics, 84(4)1913-1931--&gt;{{cite Q|Q125248630}} 
* &lt;!-- Katie Hawkinson (2024-03-27) &quot;Bernie Sanders says Israel is ‘becoming a religious fundamentalist country’&quot;, Independent--&gt;{{cite Q|Q125250312}}
* &lt;!-- Hulse (2015) Florida Ex-Senator Pursues Claims of Saudi Ties to Sept. 11 Attacks--&gt;{{cite Q|Q65002265}}
* &lt;!-- Johnson v. Parson (2024) --&gt;{{cite Q|Q125559456}}
* &lt;!-- Jake Johnson (2021) &quot;Ilhan Omar to Joe Biden: Pardon Drone Whistleblower Daniel Hale&quot;, Common Dreams--&gt;{{cite Q|Q125571612}}
* &lt;!-- Kahneman and Klein (2009) Conditions for intuitive expertise: a failure to disagree--&gt;{{cite Q|Q35001791}}
* &lt;!-- Mahsa Kaviani, Lily Yuanzhi Li, and Hosein Maleki (2022) Media, Partisan Ideology, and Corporate Social Responsibility--&gt;{{cite Q|Q121493540}}
* Rami G Khouri (2024-04-02) &quot; Watching the watchdogs: Israel’s attacks on journalists are backfiring&quot;, Al Jazeera English--&gt;{{cite Q|Q125274545}} 
* &lt;!-- Mary Elizabeth King (2007) A quiet revolution: the first Palestinian Intifada and nonviolent resistance (Bold Type Books)--&gt;{{cite Q|Q123299945}}
* &lt;!-- Mary E. King (2009) &quot;Palestinian civil resistance against Israeli military occupation&quot;, ch. 10, pp. 131-155, in Maria Stephan, ed. (2009) Civilian Jihad: Nonviolent struggle, democratization, and governance in the Middle East (Macmillan)--&gt;{{cite Q|Q108909718}}
*&lt;!-- Phillip Knightley (2004) The First Casualty, 5th ed. (Johns Hopkins U. Pr.)--&gt;{{cite Q|Q118227485}}
* &lt;!--Nikita Lalwani (2022-12-06) &quot;Rich people who own newspapers can shift elections. Israel shows how.&quot;, Washington Post--&gt;{{cite Q|Q125303071}}
* &lt;!-- Tovah Lazaroff (2022-01-19/20) &quot;World silent on Palestinian ‘terror attacks with rocks,’ Erdan tells UNSC&quot;, Jerusalem Post--&gt;{{cite Q|Q125250960}}
* &lt;!-- Likud Party (1977) Original Party Platform--&gt;{{cite Q|Q125558709}}
* &lt;!-- James Mackenzie and Nidal Al-Mughrabi (2024-04-05) Six months into Gaza war, Israel faces deepening isolation--&gt;{{cite Q|Q125352433}}
* &lt;!-- Robert W. McChesney (2004) The problem of the media (Monthly Review Press)--&gt;{{cite Q|Q7758439}}
* &lt;!-- Robert W. McChesney; John Nichols (2021). &quot;The Local Journalism Initiative: a proposal to protect and extend democracy&quot;. Columbia Journalism Review, 30 November 2021 --&gt;{{cite Q|Q109978060}}
* &lt;!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) --&gt;{{cite Q|Q109978337}}
* &lt;!-- H. R. McMaster (2020). Battlegrounds: The Fight to Defend the Free World (HarperCollins)--&gt;{{cite Q|Q104774898}}
* &lt;!-- Antonela Miho (2020) Small Screen, Big Echo? Estimating the Political Persuasion of Local Television News Bias using Sinclair Broadcast Group as a Natural Experiment--&gt;{{cite Q|Q120895467}}
* &lt;!-- E. D. Morel (1916-07) Truth and the War (National Labor Press)--&gt;{{cite Q|Q125251272}}
* &lt;!-- Yousef Munayyer (2011) &quot;Palestine’s Hidden History of Nonviolence&quot;, Foreign Policy--&gt;{{cite Q|Q125419587}}
* &lt;!-- Yoram Peri (1993), Israel: Conscientious Objection in a Democracy under Siege, pp. 146–157, ch. 12 in Charles Moskos; John Whiteclay Chambers II, eds. (1993). The New Conscientious Objection: From Sacred to Secular Resistance (Oxford U. Pr.)--&gt;{{cite Q|Q125141486|chapter=unset|volume = Moscos and Chambers (eds.) The New Conscientious Objection: From Sacred to Secular Resistance}}
* &lt;!-- Yoram Peri (1996) Afterword in&quot; The Rabin Memoirs, Expanded Edition--&gt;{{cite Q|Q125500087|author=Yoram Peri|title=''&quot;Afterword&quot; in'' The Rabin Memoirs}}
* &lt;!-- Yoram Peri (2012) &quot;The impact of national security on the development of media systems: The case of Israel&quot;, ch. 2 in Comparing media systems beyond the Western world--&gt;{{cite Q|Q125350983|chapter=unset|volume = Hallin and Mancini (eds.) Comparing media systems beyond the Western world}}
* &lt;!--Yoram Peri (2020) &quot;The Widening Military–political Gap in Israel&quot;, SWP Comment--&gt;{{cite Q|Q125364273}}
* &lt;!--Jeremy Pressman (2017) &quot;Throwing stones in social science: Non-violence, unarmed violence, and the first intifada&quot;, Cooperation and Conflict--&gt;{{cite Q|Q125419819}}
* &lt;!-- Ray Raphael (2002) The First American Revolution (he New Press)--&gt;{{cite Q|Q59420225}}
* &lt;!-- Reuters (2023-10-25) Israel to amend budget, Gaza war direct cost at $246 mln daily--&gt;{{cite Q|Q124336510|author=Reuters}}
* &lt;!-- Reuters (2024-03-29) &quot;Israel hits Syria in heaviest raid on Iran proxies in months--&gt;{{cite Q|Q125308182|author=Reuters}}
* &lt;!-- Jeremy Sharon (2024-03-20). &quot;Livni testifies at Netanyahu corruption trial she was surprised by PM's sudden change of view on newspaper law&quot;, The Times of Israel--&gt;{{cite Q|Q125145096}} 
* &lt;!--Gene Sharp (1973) The Politics of Nonviolent Action (3 vols., Porter Sargent)--&gt;{{cite Q|Q7757693}}
* &lt;!--Avi Shlaim (2014) The Iron Wall: Israel and the Arab World, 2nd ed. (Penguin)--&gt;{{cite Q|Q124213243}}
* &lt;!--SIPRI Military Expenditure Database--&gt;{{cite Q|Q125356573|date=2024-04-05|author=SIPRI}} 
* &lt;!-- South Africa v. Israel (2023)--&gt;{{cite Q|Q124208778}}
* &lt;!-- Rebecca Shimoni Stoil (2014) Adelson: Palestinians an invented people out to destroy Israel--&gt;{{cite Q|Q125616092}}
* &lt;!-- Phil Tetlock and Gardner (2015) Superforecasting: The Art and Science of Prediction--&gt;{{cite Q|Q21203378}}
* &lt;!--UN Basic Data Selection--&gt;{{cite Q|Q120485981|date=2023-07-06|title=2021 GDP (nominal)|author=UN Basic Data Selection}}
* &lt;!-- Siva Vaidhyanathan (12 June 2018). Antisocial Media: How Facebook Disconnects Us and Undermines Democracy (Oxford University Press)--&gt;{{cite Q|Q56027099}}
* &lt;!-- Tracy Wilkinson and Marcus Yam (2024-04-03) News Analysis: Netanyahu faces massive protests in Israel. Could the war in Gaza be his undoing?--&gt;{{cite Q|Q125352382}}
* &lt;!-- Ian Williams (2022-03) &quot;Hasbara and a Stone: Israel’s Ambassador Brings Both to the U.N.&quot;, Washington Report on Middle Eastern Affairs--&gt;{{cite Q|Q125251295}}
* &lt;!--Gadi Wolfsfeld, Rami Khouri, Yoram Peri (2002) &quot;News About the Other in Jordan and Israel: Does Peace Make a Diﬀerence?&quot;, Political Communication, 19:189–210,--&gt;{{cite Q|Q125474891}}
* &lt;!-- Shoshana Zuboff (2018) The Age of Surveillance Capitalism: The Fight for a Human Future at the New Frontier of Power--&gt;{{cite Q|Q75804726}}

== Notes ==
{{reflist}}
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[[Category:News]] 
[[Category:Original research]]
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[[Category:Media]]
[[Category:Freedom and abundance]]
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[[Category:News]]
[[Category:Corruption]]
[[Category:Democracy]]
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[[Category:State of Palestine]]</text>
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:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].''  

::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' 

== Abstract == 

This article evaluates how the world might be different if the Palestine Liberation Organization (PLO), founded in 1964, had sought a redress of grievances through nonviolence rather than violence. This analysis rests on a summary of research comparing the relative effectiveness of violence and nonviolence and the role of the media in conflict. It concludes with four suggestions for ending the cycle of violence and building a better future for all: (1) Demand equal protection of the laws. (2) Limit &quot;state secrets privilege&quot; to make it harder for governments to deny equal protection and lie about it with impunity. (3) Support training in nonviolence for all. (4) Citizen-directed subsidies for local news nonprofits to make it harder for major media to encourage their audiences to support counterproductive actions.

== Introduction == 

How might the world be different if the [[w:Palestine Liberation Organization|Palestine Liberation Organization]] (PLO), founded in 1964, had been committed to nonviolence, following [[w:Mahatma Gandhi|Gandhi]], [[w:Martin Luther King Jr.|King]], and [[w:Abdul Ghaffar Khan|Badshah Khan]] rather than [[w:George Washington|George Washington]] and [[w:Fidel Castro|Fidel Castro]]?

Nothing can be said about this for certain, except that the world would be different. However, careful study of history suggests that the world would most likely be better for virtually all Jews and Palestinians.&lt;ref&gt;This might be regarded as '[[w:counterfactual history| counterfactual history]]&quot;, which, especially in analyses like the present, invites people to consider the implications of alternative approaches to problems in light of research on human behavior and political economy.&lt;/ref&gt; 

This should ''NOT'' be construed as a criticism of [[w:Yassir Arafat|Yassir Arafat]] nor of anyone who supported the PLO nor any other organization that has adopted violent tactics such as [[w:Hamas|Hamas]] since 2023-10-07: They were following the example of [[w:George Washington|George Washington]]. How could they go wrong? 

Our answer to this apparent contradiction, discussed briefly below, is that few of the violent revolutions since 1776 have had the success attributed to the American Revolution, because the subsequent success of the US was achieved ''in spite of'', rather than because of, the violence of the American Revolution. The traditional narrative of the American Revolution has been written to please people who control most of the money for the media -- to the detriment of everyone else. Over 50 percent of adult white males could vote before the revolution, and the violence of the revolution did not change that, as discussed below. 

Similarly, the nonviolence of the [[w:First Intifada|First Intifada]] led to the election of [[w:Yitzhak Rabin|Yitzhak Rabin]] as Prime Minister of Israel on a platform of negotiating with Palestinians. That led to the [[w:Oslo Accords|Oslo Accords]] and the current [[w:State of Palestine|State of Palestine]]. We claim that if the Palestinians had maintained nonviolent discipline, the two-state solution promised at Oslo would likely have worked to benefit all.

Few supporters of Israel have any substantive understanding of the extent of the mistreatment of Palestinians by the Israeli military and settlers. The nonviolence of the First Intifada convinced enough Israeli voters that they could live in peace with Palestinians that Yitzhak Rabin won an election in 1992 to become Prime Minister of Israel on a platform of negotiating with Palestinians. Most nonviolent campaigns have produced similar results, as discussed below. 

Tragically, subsequent violence by both sides has created obstacles to honest consideration by each of how their opposition perceives them. Palestinians during the First Intifada and since have seen throwing rocks as relatively nonviolent. That is clearly not how most supporters of Israel have perceived that. In 2022 the Israeli ambassador to United Nations [[w:Gilad Erdan|Gilad Erdan]] complained that the world has been silent in the face of Palestinian “terror attacks with rocks” against Israelis, as he held up a rock the size of a brick. He noted that a rock like that could kill someone in a car speeding along a highway.&lt;ref&gt;Erdan's complaint was reported seriously by Lazeroff (2022) in the ''Jerusalem Post'', but was ridiculed by Willliams (2022) in the ''Washington Report on Middle Eastern Affairs''. Pressman (2017) said that throwing rocks should be considered &quot;unarmed violence&quot;.&lt;/ref&gt; 

This suggests that the nonviolence of the First Intifada, discussed below, might have been more effective if Palestinians had not thrown rocks: The shift in Israeli public opinion that got Yitzhak Rabin elected as Prime Minister would likely have been greater, and the international pressure on Israel would also likely have been greater. 

A vigorous commitment to nonviolence has worked in the past, even within the conflict between Jews and Palestinians. It seems to offer the only realistic prospect for breaking the cycle of violence and building a better future for both Palestinians and Jews.  

{{Blockquote|text=
''Oh, would some Power the gift give us''

''To see ourselves as others see us!''

''It would from many a blunder free us''
|multiline=yes
|author = [[w:Robert Burns|Robert Burns (1786)]]
|title = [[w:To a Louse|''To a Louse'']]}}

== Research comparing violence and nonviolence == 

Twenty-first century research can help us estimate the probability distribution of alternative outcomes in violent and nonviolent conflict. Most relevant in this regard is the inventory of all the major violent and nonviolent governmental change efforts of the twentieth century compiled by Erica Chenoweth and Maria Stephan (2011). They identified over 200 violent revolutions and over 100 nonviolent campaigns, each of which attracted over 1,000 people at some point. 53 percent of the nonviolent campaigns were successful while only 25 percent of the violent revolutions were. 

[[File:Democratization 1 year after vs. 1 year before twentieth century revolutions.svg|thumb|upright=2|Figure 1. Democratization 1 year after (vertical scale) vs. 1 year before (horizontal scale) the end of twentieth century revolutions]]

Probably more important than the official success rate is the impact on democracy: Chenoweth and Stephan (2011) found that on average, nonviolent campaigns ''improved'' the level of democracy, while violent revolutions had no statistically significant impact on democracy. This was true whether the campaigns won or lost. The gains for democracy tended to be greater among the nonviolent campaigns that won than among those that lost. However, even the nonviolent campaigns that lost on average pushed their governments to be more democratic, to share power more broadly; see Figure 1.

Similarly, Chenoweth and Schock (2015) noted that the presence of a &quot;[[w:radical flank effect|radical flank]]&quot;, contemporary violence pursuing similar aims, tended to ''reduce'' the probability of success. See also Chenoweth (2016).

== The nonviolence of the First Intifada == 

The NAVCO 1.1 dataset&lt;ref&gt;Chenoweth (2019a).&lt;/ref&gt; compiled by Chenoweth and Stephan includes five campaigns in Palestine or involving Palestinians: 

# &quot;'''[[w:1936–1939 Arab revolt in Palestine|Palestinian Arab Revolt]]'''&quot; in Palestine against &quot;Pro-Jewish British policies&quot; 1936-1939 coded as violent with limited success but with no change in Polity IV scores. 
# &quot;'''[[w:Mandatory Palestine#Beginning of Zionist Insurgency|Jewish resistance]]'''&quot; in &quot;Palestinian Territories&quot; against &quot;British occupation&quot; 1945-1948 coded as a violent success with no change in Polity IV scores.&lt;ref&gt;There are minor differences between how this is coded in NAVCO 1.1 and the description found in Wikipedia on 2024-03-31. For example, the section on &quot;[[w:Mandatory Palestine#Beginning of Zionist insurgency|Beginning of Zionist insurgency]]&quot; in the Wikipedia article on &quot;[[w:Mandatory Palestine|Mandatory Palestine]]&quot; mentions the assassination of &quot;Lord Moyne in Cairo&quot; 1944-11-06, while Chenoweth and Stephan (2011) coded this campaign as starting in 1945, not 1944. This discrepancy might be explained, as Chenoweth and Stephan only included cases where they &quot;were certain that more than 1,000 people were actively participating in the struggle, based on various reports.&quot; Chenoweth and Stephan may not have been able to document &quot;more than 1,000 people&quot; in that struggle prior to 1945. See also Chenoweth (2019b).&lt;/ref&gt; Chenoweth and Stephan (2011, p. 304) reported that only &quot;three successful violent insurgencies were succeeded by democratic regimes: the National Liberation Army’s 1948 victory in Costa Rica, the Jewish resistance in British-occupied Palestine, and the 1971 Bengali self-determination campaign against Pakistan. However, these instances represent only three cases out of fifty-five successful insurgencies in the twentieth century. They are as rare as authoritarian regimes that succeed victorious nonviolent campaigns. This variation points to a potentially fruitful avenue of future research&quot;, such as experiments suggested below in the section on, &quot;Implications for the future&quot;. 
# &quot;'''[[w:Black September|Palestinian activists]]'''&quot; in Jordan violently contesting &quot;Jordanian rule&quot; in 1970 coded as a failure with a modest decline from (-9) to (-10) in Polity IV scores, shifting Jordan to the most authoritarian point on the Polity IV scale.&lt;ref&gt;This doubtless refers to &quot;[[w:Black September|Black September]]&quot;, which Wikipedia reports as having run from 1970-09-06 to 1971-07-23, while NAVCO 1.1 codes both the beginning and end as 1970. This difference seems negligible for present purposes.&lt;/ref&gt; 
# '''The [[w:First Intifada|[First] Intifada]]''' in Palestine against &quot;Israeli occupation&quot; 1987-1990, coded as a partial success from nonviolence but with no change in Polity IV scores.&lt;ref&gt;The end date for the &quot;Intifada&quot; in NAVCO 1.1 is different from the description in Chenoweth and Stephan (2011), whose chapter 5 is titled, &quot;The First Palestinian Intifada, 1987-1992&quot;. The end date in the corresponding Wikipedia article was 1993-09-13 (when checked 2024-03-31), different from both the end dates in NAVCO 1.1 and Chenoweth and Stephan. However, it seems that these differences can be safely ignored for present purposes. The coding of this campaign as &quot;nonviolent&quot; and a &quot;partial success&quot; is consistent with its contributions to the Oslo Accords and its subsequent failure to achieve the two-state solution promised by Oslo. The appearance of Palestinian violence late in that campaign is consistent with the discussion of the impact of a &quot;radical flank&quot; by Chenoweth and Schock (2015) and Chenoweth (2016).&lt;/ref&gt;
# '''The longer violent &quot;Palestinian Liberation&quot; campaign''' (1973- ) against &quot;Israeli occupation&quot; beginning in 1973 and still ongoing in 2006, coded as a failure with no change in Polity IV scores. 

:''The nonviolence of the First Intifada did more to move Israeli public opinion to believe that they could live in peace and harmony with Palestinians than anything else Palestinians have done since the 1917 [[w:Balfour Declaration|Balfour Declaration]]'', at least according to the literature that we've found credible. 

When the First Intifada began, Yitzhak Rabin was Israel's Minister of Defense. He could see that the nonviolence could not be suppressed with massive counter violence for two reasons: 
# Excessive violence against nonviolent demonstrators generated bad press that was actually moving Israeli&lt;ref&gt;Peri (2012) described how Israeli public opinion towards Palestinians softened as Palestinian terrorist attacks receded into history and hardened in response to violence targeting Jews. On p. 23, he said, “In the 1990s -- during the peace process, which made it appear that the era of warfare was at an end and that Israel was becoming a postwar society -- the professional autonomy of the media grew, and journalists adopted a more critical stance. However, the failure of the peace talks in the summer of 2000 and the outbreak of the second Intifada with its suicide attacks aimed at the heart of the civilian population led to a serious retreat ... . State agencies and the public even more so again exerted pressure for media reorientation, demanding that the media restrain its criticism and circle the wagons.&quot;&lt;/ref&gt; and international opinion.&lt;ref&gt;King (2007).&lt;/ref&gt;
# Rabin knew that ''he could not count on soldiers to follow orders'' if they perceived their orders as out of proportion to the provocations.&lt;ref&gt;Peri (1993) reported that, &quot;The Palestinian uprising in the Israeli-occupied territories that began in December 1987 poses challenges of an unprecedented nature and difficulty for Israeli society. One of those challenges comes in the form of a conscientious objection to perform military service. ... At the same time, however, some one hundred officers and noncommissioned soldiers have been tried and jailed for refusing to perform military service in the West Bank and Gaza Strip. In addition to them, several thousands are in a gray area of refusal. These latter are not put on trial, and therefore no report about them goes to the higher military authorities or the public.&quot; Similarly, Peri (1996, p. 355) said that as the Intifada continued, Rabin &quot;had begun, in conversations with those close to him, to speak of a dimension that he would not dare to expand on publicly: that the war against the Intifada was damaging the [w:Israeli Defense Forces|IDF]'s fighting spirit, hurting army morale, and undermining the status of the status of the IDF as a people's army.&quot; Also, the Wikipedia article on &quot;[[w:Refusal to serve in the Israel Defense Forces|Refusal to serve in the Israel Defense Forces]]&quot; lists several organizations that have appeared consisting of Israelis who are refusing to serve in occupied territories, e.g., in Lebanon in the late 1970s and in the West Bank and Gaza in the 1980s and 1990s.&lt;/ref&gt; 

Early in the Intifada, he had told his soldiers to shoot to wound, in the legs and feet. As the nonviolence and negative press continued, he issued clubs and ordered soldiers to beat people, breaking bones.&lt;ref&gt;Munayyer (2011).&lt;/ref&gt; Before the Intifada, Rabin had not wanted to talk with Palestinians, saying, &quot;There was no point&quot;, because they always had to check with King Hussein of Jordan or President Mubarak of Egypt or President Assad of Syria.&lt;ref&gt;Peri (1996, p. 353).&lt;/ref&gt; That changed with the Intifada, because the Palestinians &quot;proved that for the first time in their history, they had decided to take charge of their fate.&quot;&lt;ref&gt;Peri (1996, p. 356).&lt;/ref&gt;

When the nonviolence continued, Rabin ran for Prime Minister on a platform of negotiating with Palestinians. He became Prime Minister in 1992 ''and was reportedly pleased when his staff told him he would not have to negotiate with leaders of the nonviolence.''&lt;ref&gt;Shlaim (2014, p. 533): &quot;Rabin’s conversion to the idea of a deal with the PLO was clinched by four evaluations ... .First ... a settlement with Syria was attainable but only at the cost of complete Israeli withdrawal from the Golan Heights. Second ... the local Palestinian leadership had finally been neutralized. Third ... Arafat’s dire situation, and possible imminent collapse, made him the most convenient interlocutor ... . Fourth ... the impressive progress achieved through the Oslo channel. Other reports that reached Rabin during this period pointed to an alarming growth in the popular following of Hamas and Islamic Jihad in the occupied territories [which] stressed to him the urgency of finding a political solution&quot;. See also King (2007, ch. 12).&lt;/ref&gt; Israeli leaders were desperate. They sent in ''[[w:agent provocateur|agents provocateurs]]'', who were exposed and neutralized until Israel expelled 481 leaders of the nonviolence and arrested between 57,000 and 120,000 others. Finally Israel got the violence they needed to justify overwhelming counter violence. That Palestinian violence also brought the ultra-Zionist right wing parties back to power in Israel.&lt;ref&gt;Mary Elizabeth King (2007, 2009) and Wikipedia, [[w:First Intifada|First Intifada]], accessed 2023-03-35.&lt;/ref&gt;

==Violence and nonviolence in the American Revolution== 

The &quot;[[w:Age of Revolution|Age of Revolution]]&quot; (1765-1849, including the French Revolution and the Latin American revolutions of the nineteenth century), plus the [[w:Russian Revolution|Russian]],&lt;ref&gt;There were actually two Russian Revolutions in 1917, the &quot;[[w:February Revolution|February]]&quot; and &quot;[[w:October Revolution|October]]&quot; Revolutions. The first was mostly nonviolent, resulting from massive popular displeasure with the management of the Russian political economy by the Tsar during World War I. The second was a violent reactions of the failures of the government that replace the Tsar, leading to the Russian Civil War. This sequence of events is crudely comparable to the First Intifada, in that the potential success of each was cut short by violence, with tragic consequences. We argue that a better popular understanding of nonviolence would likely have produced better results for both with much less loss of life.&lt;/ref&gt; [[w:Chinese Communist Revolution|Chinese]],&lt;ref&gt;The Wikipedia article on &quot;[[w:Chinese Revolution|Chinese Revolution]]&quot; lists several revolutions, all violent, none enhancing democracy. The longest and most consequential was the [[w:Chinese Communist Revolution|Chinese Communist Revolution]] (1927-1949). That's the one that would come first to many people's minds. However, it was not the only one.&lt;/ref&gt; and [[w:Cuban revolutions|Cuban]] revolutions, as well as the violent post-World War II anti-colonial struggles of Africa and Asia&lt;ref&gt;See the section on &quot;[[w: Decolonization#After 1945|After 1945]]&quot; in the Wikipedia article on &quot;[[w:Decolonization|Decolonization]]&quot;, accessed 2024-03-25.&lt;/ref&gt; all, from at least some perspectives, replaced one brutal repressive system with another. Many, perhaps all, have supporters who claim that common folk benefitted from that violence. For example, Napoleon introduced the [[w:Napoleonic Code|Napoleonic Code]], which has had a major influence on the civil code in many countries around the world, including the US state of Louisiana and most of Latin America and Eurasia.&lt;ref&gt;Wikipedia, &quot;[[w:List of national legal systems|List of national legal systems]], accessed 2024-04-01.&lt;/ref&gt; However, claims that any of this violence made substantive advances for freedom and democracy, liberty and justice are at best controversial. 

The standard narrative of the American Revolution seems to suggest that the American Revolution was different from all those other attempts to emulate it: The US, according to this narrative, got freedom and democracy, liberty and justice for all from the violence of the American Revolution. 

The reality is more nuanced: The US got independence from Great Britain. However, claims that the US got more than that from the violence are controversial and largely contradicted by the available evidence. Gaughan (2022) notes that at the time of the Revolution, Great Britain was a constitutional monarchy, which was extremely unusual during a global era of autocracy. &quot;In the British Isles, only 15 to 20 percent of English men could vote. In contrast, ... [t]he rate of enfranchisement varied from colony to colony. ... [A]s many as 80 percent of men could vote in some colonies but only 50 to 60 percent in other colonies. ... During the Revolutionary era, most states expanded suffrage to at least some degree.&quot; This occurred via ''nonviolent'' democratic deliberation as the existing legislatures of the 13 rebelling colonies either wrote state constitutions or revised their colonial charters to delete reference to Great Britain. The research of Chenoweth and Stephan (2011) suggests that the advances for democracy would likely have been greater if the rebellious colonists had refused to use violence. After the [[w:Boston Tea Party|Boston Tea Party]] in 1773, Parliament ended local self government in Massachusetts. Then crowds of worried farmers largely prevented judges and others appointed the King from doing anything unless they promised to ignore the recent acts of Parliament and abide by the colonial charter, under which those judges and other officials were answerable to the locals, ''not'' to London.&lt;ref&gt;Raphael (2002).&lt;/ref&gt; If that kind of nonviolent response had been the dominant feature of the American Revolution instead of violence, the experience of Gandhi, King, Badshah Khan, and others described by Chenoweth and Stephan (2011), Chenoweth and Schock (2015), and Chenoweth (2016) suggests that the impact of the American Revolution on world history would likely have been greater.&lt;ref&gt;See also [[The Great American Paradox]] and Graves (2005).&lt;/ref&gt;

== Other research on nonviolence ==

Previous nonviolent campaigns have often succeeded by inventing new methods of protest as needs and opportunities were identifed. After an approach obtains some level of success, people with power often develop countermeasures that reduce the effectiveness of a known technique going forward.  

Gene Sharp documented 198 nonviolent tactics.&lt;ref&gt;Sharp (1973).&lt;/ref&gt; [[w:Nonviolence International|Nonviolence International]] maintains a growing database expanding Sharp's list of 198. By 2021, Nonviolence International had documented 148 more.&lt;ref&gt;Beer (2021, p. 7; 15/116 in pdf).&lt;/ref&gt;

== Role of the media in war == 

It has been said that the first casualty of war is truth.&lt;ref&gt;Knightly (2004) attributes this to US Senator Hiram Johnson in 1917. However, the consensus in multiple articles in Wikiquote seems to attribute it to [[Wikiquote:Philip Snowden, 1st Viscount Snowden|Philip Snowden]] in his introduction to Morel (1916, p. vii).&lt;/ref&gt; We suggest, however, that 

:''Truth typically dies long before anyone is killed in armed conflict.''

In most and perhaps virtually all conflicts, each party seems to believe that their actions are justified by atrocities committed by their designated enemies. 

:''Collateral damage that &quot;they&quot; commit proves to us that &quot;they&quot; are subhuman or at best criminally misled and must be resisted by any means necessary. Meanwhile, collateral damage that we commit is unfortunate but necessary -- from our perspective.''

However, to supporters of our opposition, the collateral damage that we have committed proves to them that we are subhuman or at best criminally misled and must be resisted by any means necessary. This asymmetry of perceptions is amplified by the media each party consumes: Every media organization sells changes in audience behaviors to the people who give them money. A media organization with no audience has nothing to sell. If they have an audience but displease their funders, they may not continue to have the money needed to produce the content required to retain an audience.&lt;ref name=ConfBHow2know&gt;Wikiversity, &quot;[[Confirmation bias and conflict]]&quot; and &quot;[[How can we know?]]&quot;, accessed 2024.03.26.&lt;/ref&gt;  

Media organizations everywhere mislead their audiences. They can do this easily, because everyone prefers information and sources consistent with preconceptions, a phenomenon called &quot;[[w:confirmation bias|confirmation bias]].&quot; Media organizations everywhere exploit confirmation bias to please those who control most of the money for the media.&lt;ref name = ConfB&gt;Wikiversity, &quot;[[Confirmation bias and conflict]]&quot;.&lt;/ref&gt; Wolfsfeld et al. (2002) noted that, &quot;The news media remain important agents for demonizing enemies and transforming political and military leaders into heroes. ... News is fundamentally ethnocentric, especially news about enemies ... because they threaten us.&quot; And violence attracts more attention than nonviolence. &quot;Battles are considered newsworthy, but ideas for preventing battles are not.&quot; Also, peace negotiations are usually conducted in extreme secrecy, because excessive publicity of negotiating positions could reduce the chances of success. Once an agreement is reached, the negotiators representing different parties to the conflict must then sell the agreement to their supporters. Prime Minister Rabin and King Hussein collaborated in supporting each other in selling the [[w:Israel–Jordan peace treaty|1994 Israel–Jordan peace treaty]] to their respective peoples. 

A review of relevant literature identified multiple drivers of increases in political polarization in recent decades:&lt;ref&gt;For more on this see Wikiversity, &quot;[[Information is a public good: Designing experiments to improve government]]&quot;, accessed 2024-04-01.&lt;/ref&gt;
* Increased concentration of ownership of the media, exemplified by the creation of ''[[w:Israel Hayom|Israel Hayom]]'' by [[w:Sheldon Adelson|Sheldon Adelson]] in Israel and its impact on Israeli politics (discussed below),&lt;ref&gt;Grossman et al. (2022).&lt;/ref&gt; as well as [[w:Vincent Bolloré|Bolloré]] in France,&lt;ref&gt;Cagé (2022).&lt;/ref&gt; [[w:Rupert Murdoch|Murdoch]] in Australia, the UK and the US,&lt;ref&gt;Murdoch's business focus is exemplified in the settlement Fox accepted in ''[[w:Dominion Voting Systems v. Fox News Network|Dominion v. Fox]]'': Fox admitted that they had initially reported honestly that Biden had won the 2020 US Presidential elections. However, after finding that they were losing audience to election deniers, they switched to reporting false claims about Dominion. Fox agreed to pay Dominion $787.5 million, provided they did not have to apologize for having lied to their audience. If lying about the 2020 election increased their audience and revenue by 6% in 2021, they made money, even after paying $787.5 million to Dominion. Fraud can be good business. Media executives could be fired if they lose money trying to protect democracy.&lt;/ref&gt; and Sinclair in the US.&lt;ref&gt;Ellison (2024), Kaviani et al. (2022), Miho (2020). This trend has been extended historically to include exclusive access offered by [[w:Western Union|Western Union]], founded in 1851, to the [[w:Associated Press|Associated Press]] (AP) as long as AP reporters did not criticize major corporations and monopolies and the contribution of those biases to the rise of the [[w:Robber baron|Robber baron]]s in the US in the late nineteenth century, according to McChesney (2004, pp. 35-36): &quot;[E]conomic historians regard the growth of Western Union as a major factor in the dominance of big business in American life. ... Western Union used its monopoly power to collaborate in the development of the [[w:Associated Press| Associated Press]] [AP, founded 1846], a monopoly news service run in cooperative fashion with the largest newspaper publishers. ... With exclusive access to the wires -- Western Union refused to let potential competitors use its wires -- AP became the only wire news service in the nation. ... Needless to say, [AP] invariably presented a voice that took the side of business interests.&quot;&lt;/ref&gt; Increased concentration of ownership of the media in apparently free societies make them look more like the unfree press in authoritarian regimes like Saudi Arabia, where journalists are more often incarcerated and killed.&lt;ref&gt;[[w:Assassination of Jamal Khashoggi|On 2018-10-02 Saudi government agents killed Jamal Khashoggi]], a Saudi journalist working for the Washington Post, because they did not like his reporting. Similarly, Khouri (2024) reported that, &quot;For the past six months, Israel has put a lot of effort into covering up its genocidal crimes in Gaza. One of the most brutal ways it does this is by routinely threatening, targeting and assassinating Palestinian journalists. The US-based Committee to Protect Journalists (CPJ) has reported that at least 90 Palestinian journalists have been killed since October 7 alongside two Israelis and three Lebanese. This is the highest death toll of journalists in any modern conflict that CPJ has monitored.&quot;&lt;/ref&gt; 
* The abolition of the [[w:Fairness doctrine|Fairness doctrine]] by the US [[w:Federal Communications Commission|Federal Communications Commission]] (FCC) in 1987 and the subsequent adaptation of the major media in the US to the belief systems of their increasingly distinct audiences. ([[w:Market segmentation|Segmentation]], a standard business practice, produces political polarization threatening democracy when applied to media markets, especially when control is too concentrated.) 
* The rise of the &quot;click economy&quot;, with Internet companies, especially social media like Facebook, making money from clicks, with algorithms that exploit confirmation bias in ways that increase political polarization, herding people into echo chambers in which people become increasingly convinced of the rectitude of their own positions and unknowingly increasingly ignorant of and insensitive -- and too often hostile -- to the constructed realties of people with whom they disagree.&lt;ref&gt;Carter (2021) and Wikiversity, &quot;[[Information is a public good: Designing experiments to improve government]]&quot;, accessed 2024-04-01.&lt;/ref&gt; 
* The decline in the quantity and diversity of local news, including the growth of news deserts and ghost newspapers.&lt;ref&gt;Abernathy (2020).&lt;/ref&gt; The decline in trusted local source(s) makes it easier for people to be misled by increasingly homogenized and biased corporate media and click bait,&lt;ref&gt;Darr et al. (2018, 2021), Zuboff (2018), Frenkel and Kang (2021), Vaidhyanathan (2018).&lt;/ref&gt; even threatening the national security of the US and its allies according to retired Lt. General [[w:H. R. McMaster|McMaster]],&lt;ref&gt;McMaster (2020).&lt;/ref&gt; former President Trump's second National Security Advisor.&lt;ref&gt;For a discussion of changes like these in Germany, see Floßer (2024). He discussed the [[w:Alternative for Germany|Alternative for Germany]] (Alternatif für Deutschland, AfD), a right-wing populist, political party in Germany, that insists that [[w:German collective guilt|Germany should not feel shame or guilt]] from what it did when Hitler was their leader. Floßer said the AfD generally got more votes ''in places with no local newspaper.''&lt;/ref&gt;&lt;ref&gt;For more on the decline of local news, its impact, and what to do about it, see the section on [[International Conflict Observatory#The current legal environment for Internet and other media companies amplifies political polarization and conflict|The current legal environment for Internet and other media companies amplifies political polarization and conflict]] in the Wikiversity article on [[International Conflict Observatory]].&lt;/ref&gt; 

The abolition of the US Fairness Doctrine clearly had no legal implications outside the US. However, Grossman et al. (2022) documented an apparently similar &quot;sea change in the right’s dominance of national politics&quot; in Israel that primarily benefited Benjamin Netanyahu and his Likud party following the 2007 launch of ''[[w:Israel Hayom|Israel Hayom]]'' by billionaire casino magnate [[w:Sheldon Adelson|Sheldon Adelson]]. It was distributed for free, reportedly to skirt Israel's campaign finance laws. &quot;[I]t soon became the most widely read newspaper nationally&quot;, which Grossman et al. attribute to this newspaper. Their &quot;findings highlight the immense impact the ultrarich can exert in shaping politics through media ownership.&quot;&lt;ref&gt;See also  Lalwani (2022). Similar concerns about the impact on French politics of the media empire of billionaire Vincent Bollaré are expressed in Cagé (2022).&lt;/ref&gt; This shift was described by US Senator [[w:Bernie Sanders|Bernie Sanders]], himself Jewish, saying, &quot;the Israel of today is not the Israel of … 20 to 30 years ago ... . It is a right-wing country, increasingly becoming a religious fundamentalist country where you have some of these guys in office believe that God told them they have a right to control the entire area.&quot;&lt;ref&gt;Hawkinson (2024).&lt;/ref&gt; Sander's comment is supported by Adelson's support for the claim that, &quot;the Palestinians are an invented people out to destroy Israel&quot;,&lt;ref&gt;Stoil (2014).&lt;/ref&gt; a claim similar to the Zionist trope that Palestine was &quot;[[w:A land without a people for a people without a land|a land without a people for a people without a land]]&quot;.&lt;ref&gt;The extent to which this phrase helped drive the Zionist movement of the late nineteenth and twentieth centuries is controversial. See the Wikipedia article on &quot;[[w:A land without a people for a people without a land|a land without a people for a people without a land]]&quot;, accessed 2024-04-03.&lt;/ref&gt; 

''Israel Hayom'' is often called &quot;Bibiton&quot;, which is a portmanteau of &quot;Bibi&quot;, a nickname for Benjamin Netanyahu, and &quot;iton&quot;, the Hebrew word for newspaper. Adelson was accused&lt;ref&gt;Fulbright and Surkes (2017).&lt;/ref&gt; but not charged in the on-going corruption trial against Netanyahu&lt;ref&gt;Wikipedia, &quot;[[w:Trial of Benjamin Netanyahu|Trial of Benjamin Netanyahu]]&quot;. Netanyahu and his supporters have been working to undermine the judiciary to protect themselves from the issues raised in this trial, as documented in the Wikipedia article on &quot;[[w:2023 Israeli judicial reform|2023 Israeli judicial reform]]&quot;, accessed 2024-03-27. However, the issues raised in this legal battle are not as obviously related to questions of the role of the media in conflict, the topic of the present discussion.&lt;/ref&gt; and died before he was scheduled to testify in that trial.&lt;ref&gt;Alterman (2021).&lt;/ref&gt;

A 2022 survey in Israel found that ''Israel Hayom'' had the largest weekday readership exposure of any newspapers in Israel at 31%. The second and third most popular newspapers were ''[[w:Yedioth Ahronoth|Yedioth Ahronoth]]'' with 23.9% and ''[[w:Haaretz|Haaretz]]'' with 4.7% readership exposure.&lt;ref&gt;Readership figures are from a Hebrew-language document cited in the Wikipedia article on &quot;[[w:Newspapers in Israel|Newspapers in Israel]]&quot;, accessed 2024-04-03.&lt;/ref&gt; ''Yedioth Ahronoth'' tends to support former Israeli minister [[w:Tzipi Livni|Tzipi Livni]], who has been a leading Israeli politician advocating restraining expansion of Jewish settlements on the West Bank and negotiating with Palestinians; anything like this has been vigorously opposed by the editorial policies of ''Israel Hayom''. However, her support for Palestinians had limits, as indicated by a warrant for her arrest that was reportedly issued by a British court under universal jurisdiction, following an application by lawyers acting for Palestinian victims of the [[w:Gaza War (2008–2009)|2008-2009 Gaza war]]&lt;ref&gt;Also called &quot;Operation Cast Lead&quot;.&lt;/ref&gt; for her role in that operation as Israeli Foreign Minister. (The warrant was reportedly dropped with apologies from British political leaders after her visit to the UK was canceled.)&lt;ref&gt;Black and Cobain (2009).&lt;/ref&gt;

Political polarization doubtless exists between and within different groups supporting Palestinians and Israel driven by differences in the media they consume. Those differences in perception help perpetuate the conflict to the detriment of all.

== Implications for the future == 

What might be done to break the cycle of violence that has plagued Palestine and Israel since the 1917 Balfour declaration? 

Several ideas come to mind: 

=== Equal protection of the laws === 

Can enough Palestinians publicly and effectively renounce violence that it actually moves Israeli and international public opinion enough to force Israel to provide something like equal protection of the laws? Israel was moved in that direction by the nonviolence of the First Intifada. It could happen again. 

It would be great if Palestinians and / or Israelis and their supporters took the lead, but others do not have to wait for that. 

Rep. Marjorie Taylor Greene (R-GA) has introduced several motions in the US House of Representatives to try to censure Rep. Ilhan Omar for having said, &quot;From the river to the sea, Palestine will be free.&quot; Greene claimed that phrase was a Hamas slogan and &quot;calls for genocide of all Jews.&quot;&lt;ref&gt;Bahney (2023).&lt;/ref&gt; This has created problems for several Jewish members of the Congressional Progressive Caucus, who oppose Omar's use of that phrase but support her against censure on free speech grounds.&lt;ref&gt;Grayer (2024).&lt;/ref&gt; If Omar and others could ask instead for &quot;equal protect of the laws&quot;, it would be harder for people (like Greene) to oppose and easier for Jews and others to support.&lt;ref&gt;For more background on the use and interpretation of that and similar phrases by different groups, see the Wikipedia article on &quot;[[w:From the river to the sea|From the river to the sea]]&quot;, accessed 2024-04-09.&lt;/ref&gt;

Less well known is the portion of [[w:From the river to the sea|the 1977 election manifesto of the right-wing Israeli Likud party]] that, &quot;Between the sea and the Jordan there will only be Israeli sovereignty.&quot;&lt;ref&gt;Likud Party (1977).&lt;/ref&gt; Whether or not Ilhan Omar was intending to call for genocide of Jews, the current Israeli government, whose minister leads [[w:Likud|Likud]], has been accused of ''actual'' genocide in a complaint to the [[w:International Court of Justice|International Court of Justice]].&lt;ref&gt;[[w:South Africa v. Israel (Genocide Convention)|''South Africa v. Israel'' (2023)]].&lt;/ref&gt;

Perhaps it's better to avoid phrases like that entirely and instead demand something like &quot;equal protection of the laws.&quot; That principle was promised by the [[w:Fourteenth Amendment to the United States Constitution|Fourteenth Amendment to the US Constitution]], passed in 1868 and is still far from being implemented.&lt;ref&gt;The phrase &quot;equal protection of the laws&quot; are the last 5 words of [[w:Fourteenth Amendment to the United States Constitution#Section 1: Citizenship and civil rights|Section 1 of that amendment]]. That amendment has reportedly been the most frequently litigated part of the Constitution, and Section 1 has reportedly been the most frequently litigated part of the amendment. Just such a case active as this is being written in ''Johnson v. Parson'' (2024).&lt;/ref&gt; 

Let's broaden this question: What percent of the enemies of any country are primarily a result of routine denial of equal protection of the laws by people with power? The documentation summarized here suggests that the Palestinian violence since the Oslo Accords has largely been due to mistreatment, extended incarcerations without charges or trial, destruction of property, taking property at gun point, and even murder by Israeli security forces and settlers, unreported or underreported in the major media consumed by supporters of Israel, as noted by [[w:Ofer Cassif|Ofer Cassif]], a Jewish member of the Knesset. He said that in 2011 Israeli Prime Minister Netanyahu released [[w:Yahya Sinwar|Yahya Sinwar]], the violent leader of Hamas, in a prisoner exchange, and did ''not'' release [[w:Marwan Barghouti|Marwan Barghouti]], who has been called a Palestinian Mandela. Evidently, ''nonviolence'' is a much bigger threat than violence to the 1977 Likud promise, &quot;Between the sea and the Jordan there will only be Israeli sovereignty.&quot;&lt;ref&gt;Cassif et al. (2024).&lt;/ref&gt; Similarly, there is substantial documentation of routine denial of equal protection of foreigners by the US&lt;ref&gt;The [[w:1998 United States embassy bombings|1998 United States embassy bombings]] &quot;are widely believed to have been revenge for U.S. involvement in the extradition and alleged torture of four members of Egyptian Islamic Jihad (EIJ) who had been arrested in Albania in the two months prior to the attacks for a series of murders in Egypt&quot;, according to the Wikipedia article on those bombings, accessed 2024-04-22. If that's accurate, then the embassy bombings and the subsequent [[w:September 11 attacks|September 11 attacks]]  might not have occurred without US complicity in torture. And the September 11 attacks might not have occurred if the US had treated the embassy bombings as law enforcement issues. Beyond that, there is substantial documentation of US interference in foreign countries, e.g., overthrowing democratically elected governments to favor US international business interests. See, e.g., the Wikipedia article on [[w:Foreign interventions by the United States|Foreign interventions by the United States]] and references cited therein.&lt;/ref&gt; and France,&lt;ref&gt;See, e.g., the Wikipedia articles on [[w:Françafrique|Françafrique]] and [[w:François-Xavier Verschave|François-Xavier Verschave]] (accessed 2024-04-22), whose books titled &quot;''Françafrique'' (1999) and ''Noir silence'' (2020) have become standard works for anyone interested in the Rwandan genocide specifically, and generally the dissimulated policies followed by the French Republic in former colonies.&quot;&lt;/ref&gt; to name only two countries for which substantial documentation is available. For example, the [[w:September 11 attacks|suicide mass murders of September 11, 2001]] were major crimes but not acts of war by a major power capable of seriously threatening the United States. How might history since 2001 have been different if the US had treated the September 11 attacks as law enforcement issues and not an excuse to go to war? 

There is substantial documentation suggesting that the government of Afghanistan would likely have complied with a standard extradition request. Evidence is usually provided with extradition request,&lt;ref&gt;See the Wikipedia article on &quot;[[w:Extradition|extradition]]&quot;, accessed 2024-04-22, and references cited therein.&lt;/ref&gt; and the US refused to provide evidence when asking Afghanistan to extradite bin Laden. The documentation of the willingness of Afghanistan to consider such a request includes evidence that in 1998, June or July, the government of Afghanistan had agreed to extradite bin Laden to Saudi Arabia to stand trial for treason. If that extradition commitment had actually been carried out, bin Laden would have almost certainly been convicted and executed in typical Saudi justice. 

However, the transfer was set for September, because the Afghans needed time to separate bin Laden from his armed entourage. Then [[w:1998 United States embassy bombings|on August 7, the US embassies in Kenya and Tanzania were bombed]], and muslim clerics all over the world condemned those attacks as unjustified violence, reflecting ill on Islam. For 13 days. 

Then [[w:Operation Infinite Reach|the US bombed a pharmaceutical plant in Sudan and al-Qaeda training camps in Afghanistan]]. Muslim public opinion turned 180 degrees, as many concluded that bin Laden was correct: The US ''is'' an evil empire. The extradition was cancelled: Evidently, Afghan officials decided that if the US was not going to respect international law, maybe the world needs bin Laden. That thought was not restricted to Afghanistan. Donations to bin Laden ''increased'' substantially after the US retaliation for the embassy bombings. Employees of the Saudi embassy and consulates in the US began to support bin Laden's preparations for what became the suicide mass murders of September 11, 2001.

Afghan policy reportedly flipped again after the [[w:USS Cole bombing|bombing of the USS ''Cole'', 2000-10-12]]: Afghan officials agreed to inform the Clinton administration of bin Laden's whereabouts, so he could be killed by a US air strike. 

However, that plan was late in the Clinton administration. Implementation was passed to the incoming George W. Bush administration, which failed to act on this agreement before September 11, 2001, almost a year after the attack on the ''Cole''. Afghan officials seemed puzzled by the delay, reportedly offering in jest to provide fuel for the US aircraft to be used to kill bin Laden. 

In sum, before September 11, 2001, US government officials knew that the Afghan government had offered to help identify the whereabouts of bin Laden so he could be killed. They also knew that the government of Saudi Arabia had been involved in the preparations for the September 11 attacks. It was clear that the attacks were organized by a non-governmental entity that was not capable of threatening the internal security of the US. But rather than invading Saudi Arabia, and rather than treating Afghan officials with respect, Bush administration officials manipulated the international media to justify invading Afghanistan and Iraq. 

Senator [[w:Bob Graham|Bob Graham]] later said that during the hearings organized by the US House and Senate [[w:Joint Inquiry into Intelligence Community Activities before and after the Terrorist Attacks of September 11, 2001|Joint Inquiry into Intelligence Community Activities before and after the Terrorist Attacks of September 11, 2001]], the FBI went &quot;beyond just covering up ... into ... aggressive deception.&quot;&lt;ref&gt;Hulse (2015).&lt;/ref&gt; And then-President George W. Bush successfully convinced the committee to redact the results of that investigation from their December 2002 report. Those redactions became known as &quot;[[w:The 28 pages|The 28 pages]]&quot;. They were declassified over 13 years later by then-President Obama. During those 13 years, the US and allies were stampeded into invading Iraq by claims in news reports produced by reporters and editors, who should have known at the time were questionable and likely fraudulent.&lt;ref&gt;Similar media policies have driven other wars. These excesses were particularly egregious during and after [[w:World War I|World War I]], as documented by Hochschild (2022). The [[w:Espionage Act of 1917|Espionage Act of 1917]] gave the [[w:United States Postmaster General|Postmaster General]] the authority to declare as &quot;unmailable&quot; any publication that he decided did not adequately support the war effort. This effectively terminated many publications, because there was no other way to distribute publications nationally at that time (p. 61). The lack of broad discourse in the media amplified war hysteria, under which many people were persecuted, beaten, robbed, incarcerated, and even killed with impunity for peaceably assembling and petitioning for better wages and working conditions, or even for just speaking German. That continued after the war to help major capitalists suppress labor organizers.&lt;/ref&gt; 

What changes in the US political economy might reduce the ability of any administration to stampede the public, and the US Congress in particular, into supporting similar military actions without first exhausting non-military options?  

The US Congress could, for example, allow anyone anywhere to file suit in any US federal court for violations of ''equal protection'' by the US or Israel anyplace on earth.

=== Limit state secrets privilege === 

Equal protection of the laws cannot be guaranteed without limiting the ability of governments to deny equal protection in secret. In the US, this would require modifying the law governing &quot;[[w:State secrets privilege|state secrets privilege]]&quot;. 

Under past and current US secrecy practices, US government officials have provoked foreign entities to do things that were then denounced as unprovoked to justify counterproductive uses of military force. Current US government secrecy practices encourage such dangerous behaviors, according to documentation in Connelly (2023).&lt;ref&gt;See also the summary in Connelly, Samuelson, and Graves (2023).&lt;/ref&gt; The US responses to the 1998 embassy bombings and the September 11 attacks are examples, as discussed above. The 1997 report of the [[w:Moynihan Commission on Government Secrecy|Moynihan Commission on Government Secrecy]] reached conclusions similar to those of Connelly. Lying to Congress may officially be illegal, but exposing such lies has been punished severely while people who deceived Congress were rewarded, as documented with the revelations of [[w:Edward Snowden|Edward Snowden]]. Worse, [[w:Richard Barlow (intelligence analyst)|Richard Barlow]]'s career was reportedly destroyed merely for telling his managers they should not lie to Congress.  He did ''not'' expose officially classified government secrets, as had [[w:Whistleblowing|whistleblowers]] like Snowden: He came to public attention after filing suit for wrongful termination. 

[[w:Daniel Hale|Daniel Hale]] was sentenced to 45 months in prison for leaking a document that showed that during a five-month operation in Afghanistan, &quot;nearly 90% of the people killed during one five-month period ... were not the intended targets.&quot; Rep. Ilhan Omar asked President Biden to pardon Hale, because ''none of the documents released actually threatened US national security but instead &quot;shone a vital light on the legal and moral problems of the drone program and informed the public debate on an issue that has for too many years remained in the shadows.&quot;''&lt;ref&gt;Johnson (2021).&lt;/ref&gt; Similar things can be said about [[w:Daniel Ellsberg|Daniel Ellsberg]], [[w:Reality Winner|Reality Winner]], [[w:Chelsea Manning|Chelsea Manning]], [[w:Sibel Edmonds|Sibel Edmonds]], [[w:Jeffrey Alexander Sterling|Jeffrey Sterling]], and [[w:Thomas A. Drake|Thomas Drake]]. The attempted prosecution of [[w:Julian Assange|Julian Assange]] is similar but different, because Assange was never a US government employee and was rarely if ever in the US.&lt;ref&gt;See also Graves (2014, 2021).&lt;/ref&gt; 

The need for that much secrecy is further challenged by the research by Tetlock and Gardner (2015), who reported that their &quot;superforecasting&quot; teams &quot;performed about 30 percent better than the average for intelligence community analysts who could read intercepts and other secret data&quot; in forecasting problems assigned by the US Central Intelligence Agency (CIA).&lt;ref&gt;Tetlock and Gardner (2015, p. 95).&lt;/ref&gt; If important intelligence questions can be answered without government secrecy, that reinforces the conclusions of Connelly (2023) and others, that 

:* ''excessive government secrecy threatens more than advances US national security.''&lt;ref&gt;Connelly (2023); see also the summary in Connelly, Samuelson and Graves (2023).&lt;/ref&gt; 

We need government secrecy for some things, e.g., design and production technologies for sophisticated weapon systems and military plans for future operations. However, as the Moynihan Commission and Connelly reported, that's a tiny fraction of the information currently held as government secrets. Both claimed that the US would be safer and more prosperous if it limited government secrets to the things where government secrecy is really important. 

How can we do this? 

We suggest here that the US should authorize any federal judge to subpoena federal government documents that are classified as government secrets and declassify any, subject to appellate review, if the judge concludes that the public interest would be better served by declassification than by continued secrecy. The judge could issue a ruling with a rationale that would remain classified for a certain number of years. The ruling could go for or against the government. In a case like Ellsberg's, the judge could dismiss a criminal case with prejudice, which would preclude the government from filing charges again in front of a different judge. The judge could also award attorneys fees and damages.

=== Support training in nonviolence === 

As noted above, the nonviolence of the First Intifada led to the election of Yitzhak Rabin as Prime Minister of Israel on a platform of negotiating with Palestinians. This, in turn, led to the Oslo Accords and the current State of Palestine. As the time since the [[w:2023 Hamas-led attack on Israel|Hamas attacks on Israel of 2023-10-07]] has increased, more people in Israel,&lt;ref&gt;Wilkinson and Yam (2024).&lt;/ref&gt; the US and internationally have become increasingly concerned that the ferocity of Israeli attacks on Gaza (and the West Bank) seems far beyond the national security needs of Israel and out of proportion to the provocation.&lt;ref&gt;Mackenzie and Al-Mughrabi (2024).&lt;/ref&gt; Similarly, people are concerned that Israeli military attacks in Lebanon&lt;ref&gt;The Wikipedia article on [[w:2024 in Lebanon|2024 in Lebanon]] listed 8 attacks by Israel that killed at least 15 people in the first three months of 2024, when checked 2024-04-02.&lt;/ref&gt; and Syria&lt;ref&gt;Israel conducted lethal air strikes against alleged Iranian targets in Syria -- Aleppo March 29 and Damascus April 1, per Reuters (2024) and Bowen (2024); see also [[w:2024 Israeli bombing of the Iranian embassy in Damascus|2024 Israeli bombing of the Iranian embassy in Damascus]], accessed 2024-04-03.&lt;/ref&gt; threaten unnecessary expansion of the war.&lt;ref&gt;Peri (2020) claimed that Netanyahu's government had shifted more to the right in recent years, increasing the gap between military and political leaders. It could be helpful to have an update on how this split has impacted the current Israel-Hamas war and vice versa.&lt;/ref&gt; A new nonviolent movement led by Palestinians could strengthen people in Israel and elsewhere who are opposed to the current war.  

Palestinians and others could ask the US and Israel to support training in nonviolent civil disobedience for anyone interested, including designated &quot;terrorists&quot;. A demand like this should be difficult for people to oppose and could make it easier for people in many places to effectively work towards equal protection of the laws and other things. 

However, it is currently a criminal violation of the USA [[w:Patriot Act|Patriot Act]] to provide such training to anyone designated as a &quot;terrorist&quot; by the US State Department, per the Supreme Court decision in ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'' (2010). That law should be changed to ''support'' rather than ''criminalize'' such training.

How might the world be different if the US had vigorously supported rather than criminalized training Hamas and other designated &quot;terrorists&quot; in nonviolence? In particular, what are the chances that the [[w:2023 Hamas-led attack on Israel|Hamas-led attack on Israel of 2023-10-07]] would have happened if Palestinians could have seen progress in response to nonviolent protests against many of the outrages documented by the [[w:Palestinian Centre for Human Rights|Palestinian Centre for Human Rights]] and others? Currently few supporters of Israel seem to have any substantive awareness of the many outrages against Palestinians committed by people they support, because the media they find credible rarely if ever provides a balanced account of such outrages. If the US government had more openly supported such training, nonviolent protests facilitated by that training would likely have gotten better coverage in Israeli and international news. That in turn would likely have made it harder for the Israeli government to continue depriving Palestinians of equal protection of the laws so egregiously.

=== Citizen-directed subsidies for local news nonprofits === 

The Wikiversity article on &quot;[[Information is a public good: Designing experiments to improve government]]&quot; suggests randomized controlled trials to evaluate the long-term impact of citizen-directed subsidies for local news nonprofits distributed via local elections, as recommended by McChesney and Nichols (2021, 2022). They suggest distributing 0.15% of Gross Domestic Product (GDP) to local news non-profits via local elections with a limit on the maximum that any one local news organization could receive. Evidence summarized in that Wikiversity article suggests that the increases in political polarization in many countries in recent years may be due in part to a loss of local news, as advertising money has increasingly shifted to Internet companies that hire few if any journalists. 

To apply this to Israel and Palestine, we first note that their nominal GDPs in 2021 were $482 and $18 billion US$, respectively, totaling $500 billion.&lt;ref&gt;UN (2023).&lt;/ref&gt; 0.15% of that is $750 million. That may sound like a lot of money, but it's only 3.2% of the 2022 military budget of Israel of $23.4 billion.&lt;ref&gt;SIPRI (2024).&lt;/ref&gt; If such subsidies make a substantive contribution to reducing the political polarization that has driven this conflict for more than a century, it should help build broadly shared peace and prosperity for the long term for Israelis and Palestinians. If that happens, it would likely be the best investment in national security that Israel has made at least since 1948. This conjecture rests on the claims made above about news deserts, the &quot;click economy&quot;, and ''Israel Hayom''. If this works as predicted, it would also provide a shining example of new social technology that could help diffuse many other seemingly intractable conflicts around the globe.

=== Conclusions === 

We are not advocating an answer but a methodology that leading organizations have used successfully to build effective marketing campaigns worth billions of dollars: (i) Start by enunciating an objective like building broadly shared peace and prosperity while ending a cycle of violence. (ii) Brainstorm alternative approaches. (iii) Evaluate and refine them in focus groups. (iv) Test market the ones that seem most promising, and (v) go with what seems to work, (vi) while continuing to monitor results and adjusting accordingly. 

You, dear reader, may have other ideas that might be more effective than any of the things discussed herein to break the cycle of violence and build broadly shared peace and prosperity for the long term.

== Bibliography ==

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* &lt;!-- Katie Hawkinson (2024-03-27) &quot;Bernie Sanders says Israel is ‘becoming a religious fundamentalist country’&quot;, Independent--&gt;{{cite Q|Q125250312}}
* &lt;!-- Hulse (2015) Florida Ex-Senator Pursues Claims of Saudi Ties to Sept. 11 Attacks--&gt;{{cite Q|Q65002265}}
* &lt;!-- Johnson v. Parson (2024) --&gt;{{cite Q|Q125559456}}
* &lt;!-- Jake Johnson (2021) &quot;Ilhan Omar to Joe Biden: Pardon Drone Whistleblower Daniel Hale&quot;, Common Dreams--&gt;{{cite Q|Q125571612}}
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* Rami G Khouri (2024-04-02) &quot; Watching the watchdogs: Israel’s attacks on journalists are backfiring&quot;, Al Jazeera English--&gt;{{cite Q|Q125274545}} 
* &lt;!-- Mary Elizabeth King (2007) A quiet revolution: the first Palestinian Intifada and nonviolent resistance (Bold Type Books)--&gt;{{cite Q|Q123299945}}
* &lt;!-- Mary E. King (2009) &quot;Palestinian civil resistance against Israeli military occupation&quot;, ch. 10, pp. 131-155, in Maria Stephan, ed. (2009) Civilian Jihad: Nonviolent struggle, democratization, and governance in the Middle East (Macmillan)--&gt;{{cite Q|Q108909718}}
*&lt;!-- Phillip Knightley (2004) The First Casualty, 5th ed. (Johns Hopkins U. Pr.)--&gt;{{cite Q|Q118227485}}
* &lt;!--Nikita Lalwani (2022-12-06) &quot;Rich people who own newspapers can shift elections. Israel shows how.&quot;, Washington Post--&gt;{{cite Q|Q125303071}}
* &lt;!-- Tovah Lazaroff (2022-01-19/20) &quot;World silent on Palestinian ‘terror attacks with rocks,’ Erdan tells UNSC&quot;, Jerusalem Post--&gt;{{cite Q|Q125250960}}
* &lt;!-- Likud Party (1977) Original Party Platform--&gt;{{cite Q|Q125558709}}
* &lt;!-- James Mackenzie and Nidal Al-Mughrabi (2024-04-05) Six months into Gaza war, Israel faces deepening isolation--&gt;{{cite Q|Q125352433}}
* &lt;!-- Robert W. McChesney (2004) The problem of the media (Monthly Review Press)--&gt;{{cite Q|Q7758439}}
* &lt;!-- Robert W. McChesney; John Nichols (2021). &quot;The Local Journalism Initiative: a proposal to protect and extend democracy&quot;. Columbia Journalism Review, 30 November 2021 --&gt;{{cite Q|Q109978060}}
* &lt;!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) --&gt;{{cite Q|Q109978337}}
* &lt;!-- H. R. McMaster (2020). Battlegrounds: The Fight to Defend the Free World (HarperCollins)--&gt;{{cite Q|Q104774898}}
* &lt;!-- Antonela Miho (2020) Small Screen, Big Echo? Estimating the Political Persuasion of Local Television News Bias using Sinclair Broadcast Group as a Natural Experiment--&gt;{{cite Q|Q120895467}}
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* &lt;!-- Yousef Munayyer (2011) &quot;Palestine’s Hidden History of Nonviolence&quot;, Foreign Policy--&gt;{{cite Q|Q125419587}}
* &lt;!-- Yoram Peri (1993), Israel: Conscientious Objection in a Democracy under Siege, pp. 146–157, ch. 12 in Charles Moskos; John Whiteclay Chambers II, eds. (1993). The New Conscientious Objection: From Sacred to Secular Resistance (Oxford U. Pr.)--&gt;{{cite Q|Q125141486|chapter=unset|volume = Moscos and Chambers (eds.) The New Conscientious Objection: From Sacred to Secular Resistance}}
* &lt;!-- Yoram Peri (1996) Afterword in&quot; The Rabin Memoirs, Expanded Edition--&gt;{{cite Q|Q125500087|author=Yoram Peri|title=''&quot;Afterword&quot; in'' The Rabin Memoirs}}
* &lt;!-- Yoram Peri (2012) &quot;The impact of national security on the development of media systems: The case of Israel&quot;, ch. 2 in Comparing media systems beyond the Western world--&gt;{{cite Q|Q125350983|chapter=unset|volume = Hallin and Mancini (eds.) Comparing media systems beyond the Western world}}
* &lt;!--Yoram Peri (2020) &quot;The Widening Military–political Gap in Israel&quot;, SWP Comment--&gt;{{cite Q|Q125364273}}
* &lt;!--Jeremy Pressman (2017) &quot;Throwing stones in social science: Non-violence, unarmed violence, and the first intifada&quot;, Cooperation and Conflict--&gt;{{cite Q|Q125419819}}
* &lt;!-- Ray Raphael (2002) The First American Revolution (he New Press)--&gt;{{cite Q|Q59420225}}
* &lt;!-- Reuters (2023-10-25) Israel to amend budget, Gaza war direct cost at $246 mln daily--&gt;{{cite Q|Q124336510|author=Reuters}}
* &lt;!-- Reuters (2024-03-29) &quot;Israel hits Syria in heaviest raid on Iran proxies in months--&gt;{{cite Q|Q125308182|author=Reuters}}
* &lt;!-- Jeremy Sharon (2024-03-20). &quot;Livni testifies at Netanyahu corruption trial she was surprised by PM's sudden change of view on newspaper law&quot;, The Times of Israel--&gt;{{cite Q|Q125145096}} 
* &lt;!--Gene Sharp (1973) The Politics of Nonviolent Action (3 vols., Porter Sargent)--&gt;{{cite Q|Q7757693}}
* &lt;!--Avi Shlaim (2014) The Iron Wall: Israel and the Arab World, 2nd ed. (Penguin)--&gt;{{cite Q|Q124213243}}
* &lt;!--SIPRI Military Expenditure Database--&gt;{{cite Q|Q125356573|date=2024-04-05|author=SIPRI}} 
* &lt;!-- South Africa v. Israel (2023)--&gt;{{cite Q|Q124208778}}
* &lt;!-- Rebecca Shimoni Stoil (2014) Adelson: Palestinians an invented people out to destroy Israel--&gt;{{cite Q|Q125616092}}
* &lt;!-- Phil Tetlock and Gardner (2015) Superforecasting: The Art and Science of Prediction--&gt;{{cite Q|Q21203378}}
* &lt;!--UN Basic Data Selection--&gt;{{cite Q|Q120485981|date=2023-07-06|title=2021 GDP (nominal)|author=UN Basic Data Selection}}
* &lt;!-- Siva Vaidhyanathan (12 June 2018). Antisocial Media: How Facebook Disconnects Us and Undermines Democracy (Oxford University Press)--&gt;{{cite Q|Q56027099}}
* &lt;!-- Tracy Wilkinson and Marcus Yam (2024-04-03) News Analysis: Netanyahu faces massive protests in Israel. Could the war in Gaza be his undoing?--&gt;{{cite Q|Q125352382}}
* &lt;!-- Ian Williams (2022-03) &quot;Hasbara and a Stone: Israel’s Ambassador Brings Both to the U.N.&quot;, Washington Report on Middle Eastern Affairs--&gt;{{cite Q|Q125251295}}
* &lt;!--Gadi Wolfsfeld, Rami Khouri, Yoram Peri (2002) &quot;News About the Other in Jordan and Israel: Does Peace Make a Diﬀerence?&quot;, Political Communication, 19:189–210,--&gt;{{cite Q|Q125474891}}
* &lt;!-- Shoshana Zuboff (2018) The Age of Surveillance Capitalism: The Fight for a Human Future at the New Frontier of Power--&gt;{{cite Q|Q75804726}}

== Notes ==
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[[Category:State of Palestine]]</text>
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:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].''  

::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' 

== Abstract == 

This article evaluates how the world might be different if the Palestine Liberation Organization (PLO), founded in 1964, had sought a redress of grievances through nonviolence rather than violence. This analysis rests on a summary of research comparing the relative effectiveness of violence and nonviolence and the role of the media in conflict. It concludes with four suggestions for ending the cycle of violence and building a better future for all: (1) Demand equal protection of the laws. (2) Limit &quot;state secrets privilege&quot; to make it harder for governments to deny equal protection and lie about it with impunity. (3) Support training in nonviolence for all. (4) Citizen-directed subsidies for local news nonprofits to make it harder for major media to encourage their audiences to support counterproductive actions.

== Introduction == 

How might the world be different if the [[w:Palestine Liberation Organization|Palestine Liberation Organization]] (PLO), founded in 1964, had been committed to nonviolence, following [[w:Mahatma Gandhi|Gandhi]], [[w:Martin Luther King Jr.|King]], and [[w:Abdul Ghaffar Khan|Badshah Khan]] rather than [[w:George Washington|George Washington]] and [[w:Fidel Castro|Fidel Castro]]?

Nothing can be said about this for certain, except that the world would be different. However, careful study of history suggests that the world would most likely be better for virtually all Jews and Palestinians.&lt;ref&gt;This might be regarded as '[[w:counterfactual history| counterfactual history]]&quot;, which, especially in analyses like the present, invites people to consider the implications of alternative approaches to problems in light of research on human behavior and political economy.&lt;/ref&gt; 

This should ''NOT'' be construed as a criticism of [[w:Yassir Arafat|Yassir Arafat]] nor of anyone who supported the PLO nor any other organization that has adopted violent tactics such as [[w:Hamas|Hamas]] since 2023-10-07: They were following the example of [[w:George Washington|George Washington]]. How could they go wrong? 

Our answer to this apparent contradiction, discussed briefly below, is that few of the violent revolutions since 1776 have had the success attributed to the American Revolution, because the subsequent success of the US was achieved ''in spite of'', rather than because of, the violence of the American Revolution. The traditional narrative of the American Revolution has been written to please people who control most of the money for the media -- to the detriment of everyone else. Over 50 percent of adult white males could vote before the revolution, and the violence of the revolution did not change that, as discussed below. 

Similarly, the nonviolence of the [[w:First Intifada|First Intifada]] led to the election of [[w:Yitzhak Rabin|Yitzhak Rabin]] as Prime Minister of Israel on a platform of negotiating with Palestinians. That led to the [[w:Oslo Accords|Oslo Accords]] and the current [[w:State of Palestine|State of Palestine]]. We claim that if the Palestinians had maintained nonviolent discipline, the two-state solution promised at Oslo would likely have worked to benefit all.

Few supporters of Israel have any substantive understanding of the extent of the mistreatment of Palestinians by the Israeli military and settlers. The nonviolence of the First Intifada convinced enough Israeli voters that they could live in peace with Palestinians that Yitzhak Rabin won an election in 1992 to become Prime Minister of Israel on a platform of negotiating with Palestinians. Most nonviolent campaigns have produced similar results, as discussed below. 

Tragically, subsequent violence by both sides has created obstacles to honest consideration by each of how their opposition perceives them. Palestinians during the First Intifada and since have seen throwing rocks as relatively nonviolent. That is clearly not how most supporters of Israel have perceived that. In 2022 the Israeli ambassador to United Nations [[w:Gilad Erdan|Gilad Erdan]] complained that the world has been silent in the face of Palestinian “terror attacks with rocks” against Israelis, as he held up a rock the size of a brick. He noted that a rock like that could kill someone in a car speeding along a highway.&lt;ref&gt;Erdan's complaint was reported seriously by Lazeroff (2022) in the ''Jerusalem Post'', but was ridiculed by Willliams (2022) in the ''Washington Report on Middle Eastern Affairs''. Pressman (2017) said that throwing rocks should be considered &quot;unarmed violence&quot;.&lt;/ref&gt; 

This suggests that the nonviolence of the First Intifada, discussed below, might have been more effective if Palestinians had not thrown rocks: The shift in Israeli public opinion that got Yitzhak Rabin elected as Prime Minister would likely have been greater, and the international pressure on Israel would also likely have been greater. 

A vigorous commitment to nonviolence has worked in the past, even within the conflict between Jews and Palestinians. It seems to offer the only realistic prospect for breaking the cycle of violence and building a better future for both Palestinians and Jews.  

{{Blockquote|text=
''Oh, would some Power the gift give us''

''To see ourselves as others see us!''

''It would from many a blunder free us''
|multiline=yes
|author = [[w:Robert Burns|Robert Burns (1786)]]
|title = [[w:To a Louse|''To a Louse'']]}}

== Research comparing violence and nonviolence == 

Twenty-first century research can help us estimate the probability distribution of alternative outcomes in violent and nonviolent conflict. Most relevant in this regard is the inventory of all the major violent and nonviolent governmental change efforts of the twentieth century compiled by Erica Chenoweth and Maria Stephan (2011). They identified over 200 violent revolutions and over 100 nonviolent campaigns, each of which attracted over 1,000 people at some point. 53 percent of the nonviolent campaigns were successful while only 25 percent of the violent revolutions were. 

[[File:Democratization 1 year after vs. 1 year before twentieth century revolutions.svg|thumb|upright=2|Figure 1. Democratization 1 year after (vertical scale) vs. 1 year before (horizontal scale) the end of twentieth century revolutions]]

Probably more important than the official success rate is the impact on democracy: Chenoweth and Stephan (2011) found that on average, nonviolent campaigns ''improved'' the level of democracy, while violent revolutions had no statistically significant impact on democracy. This was true whether the campaigns won or lost. The gains for democracy tended to be greater among the nonviolent campaigns that won than among those that lost. However, even the nonviolent campaigns that lost on average pushed their governments to be more democratic, to share power more broadly; see Figure 1.

Similarly, Chenoweth and Schock (2015) noted that the presence of a &quot;[[w:radical flank effect|radical flank]]&quot;, contemporary violence pursuing similar aims, tended to ''reduce'' the probability of success. See also Chenoweth (2016).

== The nonviolence of the First Intifada == 

The NAVCO 1.1 dataset&lt;ref&gt;Chenoweth (2019a).&lt;/ref&gt; compiled by Chenoweth and Stephan includes five campaigns in Palestine or involving Palestinians: 

# &quot;'''[[w:1936–1939 Arab revolt in Palestine|Palestinian Arab Revolt]]'''&quot; in Palestine against &quot;Pro-Jewish British policies&quot; 1936-1939 coded as violent with limited success but with no change in Polity IV scores. 
# &quot;'''[[w:Mandatory Palestine#Beginning of Zionist Insurgency|Jewish resistance]]'''&quot; in &quot;Palestinian Territories&quot; against &quot;British occupation&quot; 1945-1948 coded as a violent success with no change in Polity IV scores.&lt;ref&gt;There are minor differences between how this is coded in NAVCO 1.1 and the description found in Wikipedia on 2024-03-31. For example, the section on &quot;[[w:Mandatory Palestine#Beginning of Zionist insurgency|Beginning of Zionist insurgency]]&quot; in the Wikipedia article on &quot;[[w:Mandatory Palestine|Mandatory Palestine]]&quot; mentions the assassination of &quot;Lord Moyne in Cairo&quot; 1944-11-06, while Chenoweth and Stephan (2011) coded this campaign as starting in 1945, not 1944. This discrepancy might be explained, as Chenoweth and Stephan only included cases where they &quot;were certain that more than 1,000 people were actively participating in the struggle, based on various reports.&quot; Chenoweth and Stephan may not have been able to document &quot;more than 1,000 people&quot; in that struggle prior to 1945. See also Chenoweth (2019b).&lt;/ref&gt; Chenoweth and Stephan (2011, p. 304) reported that only &quot;three successful violent insurgencies were succeeded by democratic regimes: the National Liberation Army’s 1948 victory in Costa Rica, the Jewish resistance in British-occupied Palestine, and the 1971 Bengali self-determination campaign against Pakistan. However, these instances represent only three cases out of fifty-five successful insurgencies in the twentieth century. They are as rare as authoritarian regimes that succeed victorious nonviolent campaigns. This variation points to a potentially fruitful avenue of future research&quot;, such as experiments suggested below in the section on, &quot;Implications for the future&quot;. 
# &quot;'''[[w:Black September|Palestinian activists]]'''&quot; in Jordan violently contesting &quot;Jordanian rule&quot; in 1970 coded as a failure with a modest decline from (-9) to (-10) in Polity IV scores, shifting Jordan to the most authoritarian point on the Polity IV scale.&lt;ref&gt;This doubtless refers to &quot;[[w:Black September|Black September]]&quot;, which Wikipedia reports as having run from 1970-09-06 to 1971-07-23, while NAVCO 1.1 codes both the beginning and end as 1970. This difference seems negligible for present purposes.&lt;/ref&gt; 
# '''The [[w:First Intifada|[First] Intifada]]''' in Palestine against &quot;Israeli occupation&quot; 1987-1990, coded as a partial success from nonviolence but with no change in Polity IV scores.&lt;ref&gt;The end date for the &quot;Intifada&quot; in NAVCO 1.1 is different from the description in Chenoweth and Stephan (2011), whose chapter 5 is titled, &quot;The First Palestinian Intifada, 1987-1992&quot;. The end date in the corresponding Wikipedia article was 1993-09-13 (when checked 2024-03-31), different from both the end dates in NAVCO 1.1 and Chenoweth and Stephan. However, it seems that these differences can be safely ignored for present purposes. The coding of this campaign as &quot;nonviolent&quot; and a &quot;partial success&quot; is consistent with its contributions to the Oslo Accords and its subsequent failure to achieve the two-state solution promised by Oslo. The appearance of Palestinian violence late in that campaign is consistent with the discussion of the impact of a &quot;radical flank&quot; by Chenoweth and Schock (2015) and Chenoweth (2016).&lt;/ref&gt;
# '''The longer violent &quot;Palestinian Liberation&quot; campaign''' (1973- ) against &quot;Israeli occupation&quot; beginning in 1973 and still ongoing in 2006, coded as a failure with no change in Polity IV scores. 

:''The nonviolence of the First Intifada did more to move Israeli public opinion to believe that they could live in peace and harmony with Palestinians than anything else Palestinians have done since the 1917 [[w:Balfour Declaration|Balfour Declaration]]'', at least according to the literature that we've found credible. 

When the First Intifada began, Yitzhak Rabin was Israel's Minister of Defense. He could see that the nonviolence could not be suppressed with massive counter violence for two reasons: 
# Excessive violence against nonviolent demonstrators generated bad press that was actually moving Israeli&lt;ref&gt;Peri (2012) described how Israeli public opinion towards Palestinians softened as Palestinian terrorist attacks receded into history and hardened in response to violence targeting Jews. On p. 23, he said, “In the 1990s -- during the peace process, which made it appear that the era of warfare was at an end and that Israel was becoming a postwar society -- the professional autonomy of the media grew, and journalists adopted a more critical stance. However, the failure of the peace talks in the summer of 2000 and the outbreak of the second Intifada with its suicide attacks aimed at the heart of the civilian population led to a serious retreat ... . State agencies and the public even more so again exerted pressure for media reorientation, demanding that the media restrain its criticism and circle the wagons.&quot;&lt;/ref&gt; and international opinion.&lt;ref&gt;King (2007).&lt;/ref&gt;
# Rabin knew that ''he could not count on soldiers to follow orders'' if they perceived their orders as out of proportion to the provocations.&lt;ref&gt;Peri (1993) reported that, &quot;The Palestinian uprising in the Israeli-occupied territories that began in December 1987 poses challenges of an unprecedented nature and difficulty for Israeli society. One of those challenges comes in the form of a conscientious objection to perform military service. ... At the same time, however, some one hundred officers and noncommissioned soldiers have been tried and jailed for refusing to perform military service in the West Bank and Gaza Strip. In addition to them, several thousands are in a gray area of refusal. These latter are not put on trial, and therefore no report about them goes to the higher military authorities or the public.&quot; Similarly, Peri (1996, p. 355) said that as the Intifada continued, Rabin &quot;had begun, in conversations with those close to him, to speak of a dimension that he would not dare to expand on publicly: that the war against the Intifada was damaging the [w:Israeli Defense Forces|IDF]'s fighting spirit, hurting army morale, and undermining the status of the status of the IDF as a people's army.&quot; Also, the Wikipedia article on &quot;[[w:Refusal to serve in the Israel Defense Forces|Refusal to serve in the Israel Defense Forces]]&quot; lists several organizations that have appeared consisting of Israelis who are refusing to serve in occupied territories, e.g., in Lebanon in the late 1970s and in the West Bank and Gaza in the 1980s and 1990s.&lt;/ref&gt; 

Early in the Intifada, he had told his soldiers to shoot to wound, in the legs and feet. As the nonviolence and negative press continued, he issued clubs and ordered soldiers to beat people, breaking bones.&lt;ref&gt;Munayyer (2011).&lt;/ref&gt; Before the Intifada, Rabin had not wanted to talk with Palestinians, saying, &quot;There was no point&quot;, because they always had to check with King Hussein of Jordan or President Mubarak of Egypt or President Assad of Syria.&lt;ref&gt;Peri (1996, p. 353).&lt;/ref&gt; That changed with the Intifada, because the Palestinians &quot;proved that for the first time in their history, they had decided to take charge of their fate.&quot;&lt;ref&gt;Peri (1996, p. 356).&lt;/ref&gt;

When the nonviolence continued, Rabin ran for Prime Minister on a platform of negotiating with Palestinians. He became Prime Minister in 1992 ''and was reportedly pleased when his staff told him he would not have to negotiate with leaders of the nonviolence.''&lt;ref&gt;Shlaim (2014, p. 533): &quot;Rabin’s conversion to the idea of a deal with the PLO was clinched by four evaluations ... .First ... a settlement with Syria was attainable but only at the cost of complete Israeli withdrawal from the Golan Heights. Second ... the local Palestinian leadership had finally been neutralized. Third ... Arafat’s dire situation, and possible imminent collapse, made him the most convenient interlocutor ... . Fourth ... the impressive progress achieved through the Oslo channel. Other reports that reached Rabin during this period pointed to an alarming growth in the popular following of Hamas and Islamic Jihad in the occupied territories [which] stressed to him the urgency of finding a political solution&quot;. See also King (2007, ch. 12).&lt;/ref&gt; Israeli leaders were desperate. They sent in ''[[w:agent provocateur|agents provocateurs]]'', who were exposed and neutralized until Israel expelled 481 leaders of the nonviolence and arrested between 57,000 and 120,000 others. Finally Israel got the violence they needed to justify overwhelming counter violence. That Palestinian violence also brought the ultra-Zionist right wing parties back to power in Israel.&lt;ref&gt;Mary Elizabeth King (2007, 2009) and Wikipedia, [[w:First Intifada|First Intifada]], accessed 2023-03-35.&lt;/ref&gt;

==Violence and nonviolence in the American Revolution== 

The &quot;[[w:Age of Revolution|Age of Revolution]]&quot; (1765-1849, including the French Revolution and the Latin American revolutions of the nineteenth century), plus the [[w:Russian Revolution|Russian]],&lt;ref&gt;There were actually two Russian Revolutions in 1917, the &quot;[[w:February Revolution|February]]&quot; and &quot;[[w:October Revolution|October]]&quot; Revolutions. The first was mostly nonviolent, resulting from massive popular displeasure with the management of the Russian political economy by the Tsar during World War I. The second was a violent reactions of the failures of the government that replace the Tsar, leading to the Russian Civil War. This sequence of events is crudely comparable to the First Intifada, in that the potential success of each was cut short by violence, with tragic consequences. We argue that a better popular understanding of nonviolence would likely have produced better results for both with much less loss of life.&lt;/ref&gt; [[w:Chinese Communist Revolution|Chinese]],&lt;ref&gt;The Wikipedia article on &quot;[[w:Chinese Revolution|Chinese Revolution]]&quot; lists several revolutions, all violent, none enhancing democracy. The longest and most consequential was the [[w:Chinese Communist Revolution|Chinese Communist Revolution]] (1927-1949). That's the one that would come first to many people's minds. However, it was not the only one.&lt;/ref&gt; and [[w:Cuban revolutions|Cuban]] revolutions, as well as the violent post-World War II anti-colonial struggles of Africa and Asia&lt;ref&gt;See the section on &quot;[[w: Decolonization#After 1945|After 1945]]&quot; in the Wikipedia article on &quot;[[w:Decolonization|Decolonization]]&quot;, accessed 2024-03-25.&lt;/ref&gt; all, from at least some perspectives, replaced one brutal repressive system with another. Many, perhaps all, have supporters who claim that common folk benefitted from that violence. For example, Napoleon introduced the [[w:Napoleonic Code|Napoleonic Code]], which has had a major influence on the civil code in many countries around the world, including the US state of Louisiana and most of Latin America and Eurasia.&lt;ref&gt;Wikipedia, &quot;[[w:List of national legal systems|List of national legal systems]], accessed 2024-04-01.&lt;/ref&gt; However, claims that any of this violence made substantive advances for freedom and democracy, liberty and justice are at best controversial. 

The standard narrative of the American Revolution seems to suggest that the American Revolution was different from all those other attempts to emulate it: The US, according to this narrative, got freedom and democracy, liberty and justice for all from the violence of the American Revolution. 

The reality is more nuanced: The US got independence from Great Britain. However, claims that the US got more than that from the violence are controversial and largely contradicted by the available evidence. Gaughan (2022) notes that at the time of the Revolution, Great Britain was a constitutional monarchy, which was extremely unusual during a global era of autocracy. &quot;In the British Isles, only 15 to 20 percent of English men could vote. In contrast, ... [t]he rate of enfranchisement varied from colony to colony. ... [A]s many as 80 percent of men could vote in some colonies but only 50 to 60 percent in other colonies. ... During the Revolutionary era, most states expanded suffrage to at least some degree.&quot; This occurred via ''nonviolent'' democratic deliberation as the existing legislatures of the 13 rebelling colonies either wrote state constitutions or revised their colonial charters to delete reference to Great Britain. The research of Chenoweth and Stephan (2011) suggests that the advances for democracy would likely have been greater if the rebellious colonists had refused to use violence. After the [[w:Boston Tea Party|Boston Tea Party]] in 1773, Parliament ended local self government in Massachusetts. Then crowds of worried farmers largely prevented judges and others appointed the King from doing anything unless they promised to ignore the recent acts of Parliament and abide by the colonial charter, under which those judges and other officials were answerable to the locals, ''not'' to London.&lt;ref&gt;Raphael (2002).&lt;/ref&gt; If that kind of nonviolent response had been the dominant feature of the American Revolution instead of violence, the experience of Gandhi, King, Badshah Khan, and others described by Chenoweth and Stephan (2011), Chenoweth and Schock (2015), and Chenoweth (2016) suggests that the impact of the American Revolution on world history would likely have been greater.&lt;ref&gt;See also [[The Great American Paradox]] and Graves (2005).&lt;/ref&gt;

== Other research on nonviolence ==

Previous nonviolent campaigns have often succeeded by inventing new methods of protest as needs and opportunities were identifed. After an approach obtains some level of success, people with power often develop countermeasures that reduce the effectiveness of a known technique going forward.  

Gene Sharp documented 198 nonviolent tactics.&lt;ref&gt;Sharp (1973).&lt;/ref&gt; [[w:Nonviolence International|Nonviolence International]] maintains a growing database expanding Sharp's list of 198. By 2021, Nonviolence International had documented 148 more.&lt;ref&gt;Beer (2021, p. 7; 15/116 in pdf).&lt;/ref&gt;

== Role of the media in war == 

It has been said that the first casualty of war is truth.&lt;ref&gt;Knightly (2004) attributes this to US Senator Hiram Johnson in 1917. However, the consensus in multiple articles in Wikiquote seems to attribute it to [[Wikiquote:Philip Snowden, 1st Viscount Snowden|Philip Snowden]] in his introduction to Morel (1916, p. vii).&lt;/ref&gt; We suggest, however, that 

:''Truth typically dies long before anyone is killed in armed conflict.''

In most and perhaps virtually all conflicts, each party seems to believe that their actions are justified by atrocities committed by their designated enemies. 

:''Collateral damage that &quot;they&quot; commit proves to us that &quot;they&quot; are subhuman or at best criminally misled and must be resisted by any means necessary. Meanwhile, collateral damage that we commit is unfortunate but necessary -- from our perspective.''

However, to supporters of our opposition, the collateral damage that we have committed proves to them that we are subhuman or at best criminally misled and must be resisted by any means necessary. This asymmetry of perceptions is amplified by the media each party consumes: Every media organization sells changes in audience behaviors to the people who give them money. A media organization with no audience has nothing to sell. If they have an audience but displease their funders, they may not continue to have the money needed to produce the content required to retain an audience.&lt;ref name=ConfBHow2know&gt;Wikiversity, &quot;[[Confirmation bias and conflict]]&quot; and &quot;[[How can we know?]]&quot;, accessed 2024.03.26.&lt;/ref&gt;  

Media organizations everywhere mislead their audiences. They can do this easily, because everyone prefers information and sources consistent with preconceptions, a phenomenon called &quot;[[w:confirmation bias|confirmation bias]].&quot; Media organizations everywhere exploit confirmation bias to please those who control most of the money for the media.&lt;ref name = ConfB&gt;Wikiversity, &quot;[[Confirmation bias and conflict]]&quot;.&lt;/ref&gt; Wolfsfeld et al. (2002) noted that, &quot;The news media remain important agents for demonizing enemies and transforming political and military leaders into heroes. ... News is fundamentally ethnocentric, especially news about enemies ... because they threaten us.&quot; And violence attracts more attention than nonviolence. &quot;Battles are considered newsworthy, but ideas for preventing battles are not.&quot; Also, peace negotiations are usually conducted in extreme secrecy, because excessive publicity of negotiating positions could reduce the chances of success. Once an agreement is reached, the negotiators representing different parties to the conflict must then sell the agreement to their supporters. Prime Minister Rabin and King Hussein collaborated in supporting each other in selling the [[w:Israel–Jordan peace treaty|1994 Israel–Jordan peace treaty]] to their respective peoples. 

A review of relevant literature identified multiple drivers of increases in political polarization in recent decades:&lt;ref&gt;For more on this see Wikiversity, &quot;[[Information is a public good: Designing experiments to improve government]]&quot;, accessed 2024-04-01.&lt;/ref&gt;
* Increased concentration of ownership of the media, exemplified by the creation of ''[[w:Israel Hayom|Israel Hayom]]'' by [[w:Sheldon Adelson|Sheldon Adelson]] in Israel and its impact on Israeli politics (discussed below),&lt;ref&gt;Grossman et al. (2022).&lt;/ref&gt; as well as [[w:Vincent Bolloré|Bolloré]] in France,&lt;ref&gt;Cagé (2022).&lt;/ref&gt; [[w:Rupert Murdoch|Murdoch]] in Australia, the UK and the US,&lt;ref&gt;Murdoch's business focus is exemplified in the settlement Fox accepted in ''[[w:Dominion Voting Systems v. Fox News Network|Dominion v. Fox]]'': Fox admitted that they had initially reported honestly that Biden had won the 2020 US Presidential elections. However, after finding that they were losing audience to election deniers, they switched to reporting false claims about Dominion. Fox agreed to pay Dominion $787.5 million, provided they did not have to apologize for having lied to their audience. If lying about the 2020 election increased their audience and revenue by 6% in 2021, they made money, even after paying $787.5 million to Dominion. Fraud can be good business. Media executives could be fired if they lose money trying to protect democracy.&lt;/ref&gt; and Sinclair in the US.&lt;ref&gt;Ellison (2024), Kaviani et al. (2022), Miho (2020). This trend has been extended historically to include exclusive access offered by [[w:Western Union|Western Union]], founded in 1851, to the [[w:Associated Press|Associated Press]] (AP) as long as AP reporters did not criticize major corporations and monopolies and the contribution of those biases to the rise of the [[w:Robber baron|Robber baron]]s in the US in the late nineteenth century, according to McChesney (2004, pp. 35-36): &quot;[E]conomic historians regard the growth of Western Union as a major factor in the dominance of big business in American life. ... Western Union used its monopoly power to collaborate in the development of the [[w:Associated Press| Associated Press]] [AP, founded 1846], a monopoly news service run in cooperative fashion with the largest newspaper publishers. ... With exclusive access to the wires -- Western Union refused to let potential competitors use its wires -- AP became the only wire news service in the nation. ... Needless to say, [AP] invariably presented a voice that took the side of business interests.&quot;&lt;/ref&gt; Increased concentration of ownership of the media in apparently free societies make them look more like the unfree press in authoritarian regimes like Saudi Arabia, where journalists are more often incarcerated and killed.&lt;ref&gt;[[w:Assassination of Jamal Khashoggi|On 2018-10-02 Saudi government agents killed Jamal Khashoggi]], a Saudi journalist working for the Washington Post, because they did not like his reporting. Similarly, Khouri (2024) reported that, &quot;For the past six months, Israel has put a lot of effort into covering up its genocidal crimes in Gaza. One of the most brutal ways it does this is by routinely threatening, targeting and assassinating Palestinian journalists. The US-based Committee to Protect Journalists (CPJ) has reported that at least 90 Palestinian journalists have been killed since October 7 alongside two Israelis and three Lebanese. This is the highest death toll of journalists in any modern conflict that CPJ has monitored.&quot;&lt;/ref&gt; 
* The abolition of the [[w:Fairness doctrine|Fairness doctrine]] by the US [[w:Federal Communications Commission|Federal Communications Commission]] (FCC) in 1987 and the subsequent adaptation of the major media in the US to the belief systems of their increasingly distinct audiences. ([[w:Market segmentation|Segmentation]], a standard business practice, produces political polarization threatening democracy when applied to media markets, especially when control is too concentrated.) 
* The rise of the &quot;click economy&quot;, with Internet companies, especially social media like Facebook, making money from clicks, with algorithms that exploit confirmation bias in ways that increase political polarization, herding people into echo chambers in which people become increasingly convinced of the rectitude of their own positions and unknowingly increasingly ignorant of and insensitive -- and too often hostile -- to the constructed realties of people with whom they disagree.&lt;ref&gt;Carter (2021) and Wikiversity, &quot;[[Information is a public good: Designing experiments to improve government]]&quot;, accessed 2024-04-01.&lt;/ref&gt; 
* The decline in the quantity and diversity of local news, including the growth of news deserts and ghost newspapers.&lt;ref&gt;Abernathy (2020).&lt;/ref&gt; The decline in trusted local source(s) makes it easier for people to be misled by increasingly homogenized and biased corporate media and click bait,&lt;ref&gt;Darr et al. (2018, 2021), Zuboff (2018), Frenkel and Kang (2021), Vaidhyanathan (2018).&lt;/ref&gt; even threatening the national security of the US and its allies according to retired Lt. General [[w:H. R. McMaster|McMaster]],&lt;ref&gt;McMaster (2020).&lt;/ref&gt; former President Trump's second National Security Advisor.&lt;ref&gt;For a discussion of changes like these in Germany, see Floßer (2024). He discussed the [[w:Alternative for Germany|Alternative for Germany]] (Alternatif für Deutschland, AfD), a right-wing populist, political party in Germany, that insists that [[w:German collective guilt|Germany should not feel shame or guilt]] from what it did when Hitler was their leader. Floßer said the AfD generally got more votes ''in places with no local newspaper.''&lt;/ref&gt;&lt;ref&gt;For more on the decline of local news, its impact, and what to do about it, see the section on [[International Conflict Observatory#The current legal environment for Internet and other media companies amplifies political polarization and conflict|The current legal environment for Internet and other media companies amplifies political polarization and conflict]] in the Wikiversity article on [[International Conflict Observatory]].&lt;/ref&gt; 

The abolition of the US Fairness Doctrine clearly had no legal implications outside the US. However, Grossman et al. (2022) documented an apparently similar &quot;sea change in the right’s dominance of national politics&quot; in Israel that primarily benefited Benjamin Netanyahu and his Likud party following the 2007 launch of ''[[w:Israel Hayom|Israel Hayom]]'' by billionaire casino magnate [[w:Sheldon Adelson|Sheldon Adelson]]. It was distributed for free, reportedly to skirt Israel's campaign finance laws. &quot;[I]t soon became the most widely read newspaper nationally&quot;, which Grossman et al. attribute to this newspaper. Their &quot;findings highlight the immense impact the ultrarich can exert in shaping politics through media ownership.&quot;&lt;ref&gt;See also  Lalwani (2022). Similar concerns about the impact on French politics of the media empire of billionaire Vincent Bollaré are expressed in Cagé (2022).&lt;/ref&gt; This shift was described by US Senator [[w:Bernie Sanders|Bernie Sanders]], himself Jewish, saying, &quot;the Israel of today is not the Israel of … 20 to 30 years ago ... . It is a right-wing country, increasingly becoming a religious fundamentalist country where you have some of these guys in office believe that God told them they have a right to control the entire area.&quot;&lt;ref&gt;Hawkinson (2024).&lt;/ref&gt; Sander's comment is supported by Adelson's support for the claim that, &quot;the Palestinians are an invented people out to destroy Israel&quot;,&lt;ref&gt;Stoil (2014).&lt;/ref&gt; a claim similar to the Zionist trope that Palestine was &quot;[[w:A land without a people for a people without a land|a land without a people for a people without a land]]&quot;.&lt;ref&gt;The extent to which this phrase helped drive the Zionist movement of the late nineteenth and twentieth centuries is controversial. See the Wikipedia article on &quot;[[w:A land without a people for a people without a land|a land without a people for a people without a land]]&quot;, accessed 2024-04-03.&lt;/ref&gt; 

''Israel Hayom'' is often called &quot;Bibiton&quot;, which is a portmanteau of &quot;Bibi&quot;, a nickname for Benjamin Netanyahu, and &quot;iton&quot;, the Hebrew word for newspaper. Adelson was accused&lt;ref&gt;Fulbright and Surkes (2017).&lt;/ref&gt; but not charged in the on-going corruption trial against Netanyahu&lt;ref&gt;Wikipedia, &quot;[[w:Trial of Benjamin Netanyahu|Trial of Benjamin Netanyahu]]&quot;. Netanyahu and his supporters have been working to undermine the judiciary to protect themselves from the issues raised in this trial, as documented in the Wikipedia article on &quot;[[w:2023 Israeli judicial reform|2023 Israeli judicial reform]]&quot;, accessed 2024-03-27. However, the issues raised in this legal battle are not as obviously related to questions of the role of the media in conflict, the topic of the present discussion.&lt;/ref&gt; and died before he was scheduled to testify in that trial.&lt;ref&gt;Alterman (2021).&lt;/ref&gt;

A 2022 survey in Israel found that ''Israel Hayom'' had the largest weekday readership exposure of any newspapers in Israel at 31%. The second and third most popular newspapers were ''[[w:Yedioth Ahronoth|Yedioth Ahronoth]]'' with 23.9% and ''[[w:Haaretz|Haaretz]]'' with 4.7% readership exposure.&lt;ref&gt;Readership figures are from a Hebrew-language document cited in the Wikipedia article on &quot;[[w:Newspapers in Israel|Newspapers in Israel]]&quot;, accessed 2024-04-03.&lt;/ref&gt; ''Yedioth Ahronoth'' tends to support former Israeli minister [[w:Tzipi Livni|Tzipi Livni]], who has been a leading Israeli politician advocating restraining expansion of Jewish settlements on the West Bank and negotiating with Palestinians; anything like this has been vigorously opposed by the editorial policies of ''Israel Hayom''. However, her support for Palestinians had limits, as indicated by a warrant for her arrest that was reportedly issued by a British court under universal jurisdiction, following an application by lawyers acting for Palestinian victims of the [[w:Gaza War (2008–2009)|2008-2009 Gaza war]]&lt;ref&gt;Also called &quot;Operation Cast Lead&quot;.&lt;/ref&gt; for her role in that operation as Israeli Foreign Minister. (The warrant was reportedly dropped with apologies from British political leaders after her visit to the UK was canceled.)&lt;ref&gt;Black and Cobain (2009).&lt;/ref&gt;

Political polarization doubtless exists between and within different groups supporting Palestinians and Israel driven by differences in the media they consume. Those differences in perception help perpetuate the conflict to the detriment of all.

== Implications for the future == 

What might be done to break the cycle of violence that has plagued Palestine and Israel since the 1917 Balfour declaration? 

Several ideas come to mind: 

=== Equal protection of the laws === 

Can enough Palestinians publicly and effectively renounce violence that it actually moves Israeli and international public opinion enough to force Israel to provide something like equal protection of the laws? Israel was moved in that direction by the nonviolence of the First Intifada. It could happen again. 

It would be great if Palestinians and / or Israelis and their supporters took the lead, but others do not have to wait for that. 

Rep. Marjorie Taylor Greene (R-GA) has introduced several motions in the US House of Representatives to try to censure Rep. Ilhan Omar for having said, &quot;From the river to the sea, Palestine will be free.&quot; Greene claimed that phrase was a Hamas slogan and &quot;calls for genocide of all Jews.&quot;&lt;ref&gt;Bahney (2023).&lt;/ref&gt; This has created problems for several Jewish members of the Congressional Progressive Caucus, who oppose Omar's use of that phrase but support her against censure on free speech grounds.&lt;ref&gt;Grayer (2024).&lt;/ref&gt; If Omar and others could ask instead for &quot;equal protect of the laws&quot;, it would be harder for people (like Greene) to oppose and easier for Jews and others to support.&lt;ref&gt;For more background on the use and interpretation of that and similar phrases by different groups, see the Wikipedia article on &quot;[[w:From the river to the sea|From the river to the sea]]&quot;, accessed 2024-04-09.&lt;/ref&gt;

Less well known is the portion of [[w:From the river to the sea|the 1977 election manifesto of the right-wing Israeli Likud party]] that, &quot;Between the sea and the Jordan there will only be Israeli sovereignty.&quot;&lt;ref&gt;Likud Party (1977).&lt;/ref&gt; Whether or not Ilhan Omar was intending to call for genocide of Jews, the current Israeli government, whose minister leads [[w:Likud|Likud]], has been accused of ''actual'' genocide in a complaint to the [[w:International Court of Justice|International Court of Justice]].&lt;ref&gt;[[w:South Africa v. Israel (Genocide Convention)|''South Africa v. Israel'' (2023)]].&lt;/ref&gt; Those accusations are consistent with the 1977 Lukud party platform and the comment by billionaire Sheldon Adelson, whose ''Israel Hayom'' is the leading newspaper delivered for free to anyone who wants it that, the &quot;Palestinians [are] an invented people out to destroy Israel.&quot;&lt;ref&gt;Evidence from many sources suggests that the vast majority of Palestinians are ''not'' out to destroy Israel. Past Palestinian violence can be plausibly attributed to mistreatment they have experienced.&lt;/ref&gt;

Perhaps it's better to avoid phrases like that entirely and instead demand something like &quot;equal protection of the laws.&quot; That principle was promised by the [[w:Fourteenth Amendment to the United States Constitution|Fourteenth Amendment to the US Constitution]], passed in 1868 and is still far from being implemented.&lt;ref&gt;The phrase &quot;equal protection of the laws&quot; are the last 5 words of [[w:Fourteenth Amendment to the United States Constitution#Section 1: Citizenship and civil rights|Section 1 of that amendment]]. That amendment has reportedly been the most frequently litigated part of the Constitution, and Section 1 has reportedly been the most frequently litigated part of the amendment. Just such a case active as this is being written in ''Johnson v. Parson'' (2024).&lt;/ref&gt; 

Let's broaden this question: What percent of the enemies of any country are primarily a result of routine denial of equal protection of the laws by people with power? The documentation summarized here suggests that the Palestinian violence since the Oslo Accords has largely been due to mistreatment, extended incarcerations without charges or trial, destruction of property, taking property at gun point, and even murder by Israeli security forces and settlers, unreported or underreported in the major media consumed by supporters of Israel, as noted by [[w:Ofer Cassif|Ofer Cassif]], a Jewish member of the Knesset. He said that in 2011 Israeli Prime Minister Netanyahu released [[w:Yahya Sinwar|Yahya Sinwar]], the violent leader of Hamas, in a prisoner exchange, and did ''not'' release [[w:Marwan Barghouti|Marwan Barghouti]], who has been called a Palestinian Mandela. Evidently, ''nonviolence'' is a much bigger threat than violence to the 1977 Likud promise, &quot;Between the sea and the Jordan there will only be Israeli sovereignty.&quot;&lt;ref&gt;Cassif et al. (2024).&lt;/ref&gt; Similarly, there is substantial documentation of routine denial of equal protection of foreigners by the US&lt;ref&gt;The [[w:1998 United States embassy bombings|1998 United States embassy bombings]] &quot;are widely believed to have been revenge for U.S. involvement in the extradition and alleged torture of four members of Egyptian Islamic Jihad (EIJ) who had been arrested in Albania in the two months prior to the attacks for a series of murders in Egypt&quot;, according to the Wikipedia article on those bombings, accessed 2024-04-22. If that's accurate, then the embassy bombings and the subsequent [[w:September 11 attacks|September 11 attacks]]  might not have occurred without US complicity in torture. And the September 11 attacks might not have occurred if the US had treated the embassy bombings as law enforcement issues. Beyond that, there is substantial documentation of US interference in foreign countries, e.g., overthrowing democratically elected governments to favor US international business interests. See, e.g., the Wikipedia article on [[w:Foreign interventions by the United States|Foreign interventions by the United States]] and references cited therein.&lt;/ref&gt; and France,&lt;ref&gt;See, e.g., the Wikipedia articles on [[w:Françafrique|Françafrique]] and [[w:François-Xavier Verschave|François-Xavier Verschave]] (accessed 2024-04-22), whose books titled &quot;''Françafrique'' (1999) and ''Noir silence'' (2020) have become standard works for anyone interested in the Rwandan genocide specifically, and generally the dissimulated policies followed by the French Republic in former colonies.&quot;&lt;/ref&gt; to name only two countries for which substantial documentation is available. For example, the [[w:September 11 attacks|suicide mass murders of September 11, 2001]] were major crimes but not acts of war by a major power capable of seriously threatening the United States. How might history since 2001 have been different if the US had treated the September 11 attacks as law enforcement issues and not an excuse to go to war? 

There is substantial documentation suggesting that the government of Afghanistan would likely have complied with a standard extradition request. Evidence is usually provided with extradition request,&lt;ref&gt;See the Wikipedia article on &quot;[[w:Extradition|extradition]]&quot;, accessed 2024-04-22, and references cited therein.&lt;/ref&gt; and the US refused to provide evidence when asking Afghanistan to extradite bin Laden. The documentation of the willingness of Afghanistan to consider such a request includes evidence that in 1998, June or July, the government of Afghanistan had agreed to extradite bin Laden to Saudi Arabia to stand trial for treason. If that extradition commitment had actually been carried out, bin Laden would have almost certainly been convicted and executed in typical Saudi justice. 

However, the transfer was set for September, because the Afghans needed time to separate bin Laden from his armed entourage. Then [[w:1998 United States embassy bombings|on August 7, the US embassies in Kenya and Tanzania were bombed]], and muslim clerics all over the world condemned those attacks as unjustified violence, reflecting ill on Islam. For 13 days. 

Then [[w:Operation Infinite Reach|the US bombed a pharmaceutical plant in Sudan and al-Qaeda training camps in Afghanistan]]. Muslim public opinion turned 180 degrees, as many concluded that bin Laden was correct: The US ''is'' an evil empire. The extradition was cancelled: Evidently, Afghan officials decided that if the US was not going to respect international law, maybe the world needs bin Laden. That thought was not restricted to Afghanistan. Donations to bin Laden ''increased'' substantially after the US retaliation for the embassy bombings. Employees of the Saudi embassy and consulates in the US began to support bin Laden's preparations for what became the suicide mass murders of September 11, 2001.

Afghan policy reportedly flipped again after the [[w:USS Cole bombing|bombing of the USS ''Cole'', 2000-10-12]]: Afghan officials agreed to inform the Clinton administration of bin Laden's whereabouts, so he could be killed by a US air strike. 

However, that plan was late in the Clinton administration. Implementation was passed to the incoming George W. Bush administration, which failed to act on this agreement before September 11, 2001, almost a year after the attack on the ''Cole''. Afghan officials seemed puzzled by the delay, reportedly offering in jest to provide fuel for the US aircraft to be used to kill bin Laden. 

In sum, before September 11, 2001, US government officials knew that the Afghan government had offered to help identify the whereabouts of bin Laden so he could be killed. They also knew that the government of Saudi Arabia had been involved in the preparations for the September 11 attacks. It was clear that the attacks were organized by a non-governmental entity that was not capable of threatening the internal security of the US. But rather than invading Saudi Arabia, and rather than treating Afghan officials with respect, Bush administration officials manipulated the international media to justify invading Afghanistan and Iraq. 

Senator [[w:Bob Graham|Bob Graham]] later said that during the hearings organized by the US House and Senate [[w:Joint Inquiry into Intelligence Community Activities before and after the Terrorist Attacks of September 11, 2001|Joint Inquiry into Intelligence Community Activities before and after the Terrorist Attacks of September 11, 2001]], the FBI went &quot;beyond just covering up ... into ... aggressive deception.&quot;&lt;ref&gt;Hulse (2015).&lt;/ref&gt; And then-President George W. Bush successfully convinced the committee to redact the results of that investigation from their December 2002 report. Those redactions became known as &quot;[[w:The 28 pages|The 28 pages]]&quot;. They were declassified over 13 years later by then-President Obama. During those 13 years, the US and allies were stampeded into invading Iraq by claims in news reports produced by reporters and editors, who should have known at the time were questionable and likely fraudulent.&lt;ref&gt;Similar media policies have driven other wars. These excesses were particularly egregious during and after [[w:World War I|World War I]], as documented by Hochschild (2022). The [[w:Espionage Act of 1917|Espionage Act of 1917]] gave the [[w:United States Postmaster General|Postmaster General]] the authority to declare as &quot;unmailable&quot; any publication that he decided did not adequately support the war effort. This effectively terminated many publications, because there was no other way to distribute publications nationally at that time (p. 61). The lack of broad discourse in the media amplified war hysteria, under which many people were persecuted, beaten, robbed, incarcerated, and even killed with impunity for peaceably assembling and petitioning for better wages and working conditions, or even for just speaking German. That continued after the war to help major capitalists suppress labor organizers.&lt;/ref&gt; 

What changes in the US political economy might reduce the ability of any administration to stampede the public, and the US Congress in particular, into supporting similar military actions without first exhausting non-military options?  

The US Congress could, for example, allow anyone anywhere to file suit in any US federal court for violations of ''equal protection'' by the US or Israel anyplace on earth.

=== Limit state secrets privilege === 

Equal protection of the laws cannot be guaranteed without limiting the ability of governments to deny equal protection in secret. In the US, this would require modifying the law governing &quot;[[w:State secrets privilege|state secrets privilege]]&quot;. 

Under past and current US secrecy practices, US government officials have provoked foreign entities to do things that were then denounced as unprovoked to justify counterproductive uses of military force. Current US government secrecy practices encourage such dangerous behaviors, according to documentation in Connelly (2023).&lt;ref&gt;See also the summary in Connelly, Samuelson, and Graves (2023).&lt;/ref&gt; The US responses to the 1998 embassy bombings and the September 11 attacks are examples, as discussed above. The 1997 report of the [[w:Moynihan Commission on Government Secrecy|Moynihan Commission on Government Secrecy]] reached conclusions similar to those of Connelly. Lying to Congress may officially be illegal, but exposing such lies has been punished severely while people who deceived Congress were rewarded, as documented with the revelations of [[w:Edward Snowden|Edward Snowden]]. Worse, [[w:Richard Barlow (intelligence analyst)|Richard Barlow]]'s career was reportedly destroyed merely for telling his managers they should not lie to Congress.  He did ''not'' expose officially classified government secrets, as had [[w:Whistleblowing|whistleblowers]] like Snowden: He came to public attention after filing suit for wrongful termination. 

[[w:Daniel Hale|Daniel Hale]] was sentenced to 45 months in prison for leaking a document that showed that during a five-month operation in Afghanistan, &quot;nearly 90% of the people killed during one five-month period ... were not the intended targets.&quot; Rep. Ilhan Omar asked President Biden to pardon Hale, because ''none of the documents released actually threatened US national security but instead &quot;shone a vital light on the legal and moral problems of the drone program and informed the public debate on an issue that has for too many years remained in the shadows.&quot;''&lt;ref&gt;Johnson (2021).&lt;/ref&gt; Similar things can be said about [[w:Daniel Ellsberg|Daniel Ellsberg]], [[w:Reality Winner|Reality Winner]], [[w:Chelsea Manning|Chelsea Manning]], [[w:Sibel Edmonds|Sibel Edmonds]], [[w:Jeffrey Alexander Sterling|Jeffrey Sterling]], and [[w:Thomas A. Drake|Thomas Drake]]. The attempted prosecution of [[w:Julian Assange|Julian Assange]] is similar but different, because Assange was never a US government employee and was rarely if ever in the US.&lt;ref&gt;See also Graves (2014, 2021).&lt;/ref&gt; 

The need for that much secrecy is further challenged by the research by Tetlock and Gardner (2015), who reported that their &quot;superforecasting&quot; teams &quot;performed about 30 percent better than the average for intelligence community analysts who could read intercepts and other secret data&quot; in forecasting problems assigned by the US Central Intelligence Agency (CIA).&lt;ref&gt;Tetlock and Gardner (2015, p. 95).&lt;/ref&gt; If important intelligence questions can be answered without government secrecy, that reinforces the conclusions of Connelly (2023) and others, that 

:* ''excessive government secrecy threatens more than advances US national security.''&lt;ref&gt;Connelly (2023); see also the summary in Connelly, Samuelson and Graves (2023).&lt;/ref&gt; 

We need government secrecy for some things, e.g., design and production technologies for sophisticated weapon systems and military plans for future operations. However, as the Moynihan Commission and Connelly reported, that's a tiny fraction of the information currently held as government secrets. Both claimed that the US would be safer and more prosperous if it limited government secrets to the things where government secrecy is really important. 

How can we do this? 

We suggest here that the US should authorize any federal judge to subpoena federal government documents that are classified as government secrets and declassify any, subject to appellate review, if the judge concludes that the public interest would be better served by declassification than by continued secrecy. The judge could issue a ruling with a rationale that would remain classified for a certain number of years. The ruling could go for or against the government. In a case like Ellsberg's, the judge could dismiss a criminal case with prejudice, which would preclude the government from filing charges again in front of a different judge. The judge could also award attorneys fees and damages.

=== Support training in nonviolence === 

As noted above, the nonviolence of the First Intifada led to the election of Yitzhak Rabin as Prime Minister of Israel on a platform of negotiating with Palestinians. This, in turn, led to the Oslo Accords and the current State of Palestine. As the time since the [[w:2023 Hamas-led attack on Israel|Hamas attacks on Israel of 2023-10-07]] has increased, more people in Israel,&lt;ref&gt;Wilkinson and Yam (2024).&lt;/ref&gt; the US and internationally have become increasingly concerned that the ferocity of Israeli attacks on Gaza (and the West Bank) seems far beyond the national security needs of Israel and out of proportion to the provocation.&lt;ref&gt;Mackenzie and Al-Mughrabi (2024).&lt;/ref&gt; Similarly, people are concerned that Israeli military attacks in Lebanon&lt;ref&gt;The Wikipedia article on [[w:2024 in Lebanon|2024 in Lebanon]] listed 8 attacks by Israel that killed at least 15 people in the first three months of 2024, when checked 2024-04-02.&lt;/ref&gt; and Syria&lt;ref&gt;Israel conducted lethal air strikes against alleged Iranian targets in Syria -- Aleppo March 29 and Damascus April 1, per Reuters (2024) and Bowen (2024); see also [[w:2024 Israeli bombing of the Iranian embassy in Damascus|2024 Israeli bombing of the Iranian embassy in Damascus]], accessed 2024-04-03.&lt;/ref&gt; threaten unnecessary expansion of the war.&lt;ref&gt;Peri (2020) claimed that Netanyahu's government had shifted more to the right in recent years, increasing the gap between military and political leaders. It could be helpful to have an update on how this split has impacted the current Israel-Hamas war and vice versa.&lt;/ref&gt; A new nonviolent movement led by Palestinians could strengthen people in Israel and elsewhere who are opposed to the current war.  

Palestinians and others could ask the US and Israel to support training in nonviolent civil disobedience for anyone interested, including designated &quot;terrorists&quot;. A demand like this should be difficult for people to oppose and could make it easier for people in many places to effectively work towards equal protection of the laws and other things. 

However, it is currently a criminal violation of the USA [[w:Patriot Act|Patriot Act]] to provide such training to anyone designated as a &quot;terrorist&quot; by the US State Department, per the Supreme Court decision in ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'' (2010). That law should be changed to ''support'' rather than ''criminalize'' such training.

How might the world be different if the US had vigorously supported rather than criminalized training Hamas and other designated &quot;terrorists&quot; in nonviolence? In particular, what are the chances that the [[w:2023 Hamas-led attack on Israel|Hamas-led attack on Israel of 2023-10-07]] would have happened if Palestinians could have seen progress in response to nonviolent protests against many of the outrages documented by the [[w:Palestinian Centre for Human Rights|Palestinian Centre for Human Rights]] and others? Currently few supporters of Israel seem to have any substantive awareness of the many outrages against Palestinians committed by people they support, because the media they find credible rarely if ever provides a balanced account of such outrages. If the US government had more openly supported such training, nonviolent protests facilitated by that training would likely have gotten better coverage in Israeli and international news. That in turn would likely have made it harder for the Israeli government to continue depriving Palestinians of equal protection of the laws so egregiously.

=== Citizen-directed subsidies for local news nonprofits === 

The Wikiversity article on &quot;[[Information is a public good: Designing experiments to improve government]]&quot; suggests randomized controlled trials to evaluate the long-term impact of citizen-directed subsidies for local news nonprofits distributed via local elections, as recommended by McChesney and Nichols (2021, 2022). They suggest distributing 0.15% of Gross Domestic Product (GDP) to local news non-profits via local elections with a limit on the maximum that any one local news organization could receive. Evidence summarized in that Wikiversity article suggests that the increases in political polarization in many countries in recent years may be due in part to a loss of local news, as advertising money has increasingly shifted to Internet companies that hire few if any journalists. 

To apply this to Israel and Palestine, we first note that their nominal GDPs in 2021 were $482 and $18 billion US$, respectively, totaling $500 billion.&lt;ref&gt;UN (2023).&lt;/ref&gt; 0.15% of that is $750 million. That may sound like a lot of money, but it's only 3.2% of the 2022 military budget of Israel of $23.4 billion.&lt;ref&gt;SIPRI (2024).&lt;/ref&gt; If such subsidies make a substantive contribution to reducing the political polarization that has driven this conflict for more than a century, it should help build broadly shared peace and prosperity for the long term for Israelis and Palestinians. If that happens, it would likely be the best investment in national security that Israel has made at least since 1948. This conjecture rests on the claims made above about news deserts, the &quot;click economy&quot;, and ''Israel Hayom''. If this works as predicted, it would also provide a shining example of new social technology that could help diffuse many other seemingly intractable conflicts around the globe.

=== Conclusions === 

We are not advocating an answer but a methodology that leading organizations have used successfully to build effective marketing campaigns worth billions of dollars: (i) Start by enunciating an objective like building broadly shared peace and prosperity while ending a cycle of violence. (ii) Brainstorm alternative approaches. (iii) Evaluate and refine them in focus groups. (iv) Test market the ones that seem most promising, and (v) go with what seems to work, (vi) while continuing to monitor results and adjusting accordingly. 

You, dear reader, may have other ideas that might be more effective than any of the things discussed herein to break the cycle of violence and build broadly shared peace and prosperity for the long term.

== Bibliography ==

* &lt;!-- Penny Abernathy (2020)  News Deserts and Ghost Newspapers: Will local news survive?--&gt;{{cite Q|Q100251717}} 
* &lt;!-- Eric Alterman (2021-02-08) &quot;Even in Death, Sheldon Adelson Will Keep Undermining Democracy&quot;, The Nation--&gt;{{cite Q|Q125145267}}
* &lt;!--- Jennifer Bowers Bahney (2023-11-06) &quot;Here We Go Again: Marjorie Taylor Greene Introduces New Censure Resolution Against Rashida Tlaib&quot;, Microsoft Start --&gt;{{cite Q|Q125246907}}
* &lt;!-- Ian Black and Ian Cobain (2009) &quot;British court issued Gaza arrest warrant for former Israeli minister Tzipi Livni--&gt;{{cite Q|Q125157422)}}
* &lt;!-- Revital Blumenfeld (2011-12-26) &quot;Sheldon Adelson to Birthright Group: Gingrich Is Right to Call Palestinians 'Invented People'&quot;, Haaretz--&gt;{{cite Q|Q125302285}}
* &lt;!-- Jeromy Bowen (2024-04-01) &quot;Iran accuses Israel of killing generals in Syria strike&quot;, BBC--&gt;{{cite Q|Q125308201}}
* &lt;!-- Julia Cagé (4 February 2022). Pour une télé libre : Contre Bolloré (in fr; Éditions du Seuil)--&gt;{{cite Q|Q120891998}}
* &lt;!--Timothy Carter (8 February 2021). &quot;The Click Economy: Is There a Future?&quot;, ReadWrite--&gt;{{cite Q|Q125153120}}
* &lt;!-- Cassif et al. (2024) &quot;Against genocide: A conversation with Ofer Cassif&quot;--&gt;{{cite Q|Q125559918}} 
* &lt;!-- Erica Chenoweth (2016-11-21) People are in the streets protesting Donald Trump. But when does protest actually work?, Washington Post--&gt;{{cite Q|Q125360193}}
* &lt;!-- Erica Chenoweth (2019a) NAVCO 1.1 Dataset--&gt;{{cite Q|Q125247341|date=2019a}}
* &lt;!-- Erica Chenoweth (2019b) WCRW Web Appendix.pdf--&gt;{{cite Q|Q125247400|date=2019b}}
* &lt;!-- Erica Chenoweth and Kurt Schock (2015) Do Contemporaneous Armed Challenges Affect the Outcomes of Mass Nonviolent Campaigns?, Mobilization--&gt;{{cite Q|Q83970885}}
* &lt;!-- Erica Chenoweth; Maria J. Stephan (2011). Why Civil Resistance Works: The Strategic Logic of Nonviolent Conflict (Columbia U. Pr.)--&gt;{{cite Q|Q88725216}}
* &lt;!-- Connelly (2023) The Declassification Engine: What History Reveals About America's Top Secrets--&gt;{{cite Q|Q116786691}}
* &lt;!-- Connelly, Samuelson and Graves (2023) Does US government secrecy threaten national security?--&gt;{{cite Q|Q125582094}}
* &lt;!-- Darr, Hitt, and Dunaway (2018) &quot;Newspaper Closures Polarize Voting Behavior&quot;, J. of Communication--&gt;{{cite Q|Q61311391}} 
* &lt;!-- Darr, Hitt, and Dunaway (2021)  Home Style Opinion: How Local Newspapers Can Slow Polarization--&gt;{{cite Q|Q125287538}} 
* &lt;!-- Sarah Ellison (2024-02-16) &quot;Sinclair's recipe for TV news: Crime, homelessness, illegal drugs&quot;, Washington Post--&gt;{{cite Q|Q125271605}}
* Maxim Floßer (2024-03-06) &quot;Keine Lokalzeitung -- mehr AfD&quot;, Kontext--&gt;{{cite Q|Q125287792}}
* &lt;!--Sheera Frenkel; Cecilia Kang (July 2021). An Ugly Truth: Inside Facebook's Battle for Domination (Harper)--&gt;{{cite Q|Q107560533}}
* &lt;!-- Alexander Fulbright and Sue Surkes (2017-06-26) &quot;Adelson testifies for second time in Netanyahu corruption probe&quot;, Times of Israel --&gt;{{cite Q|Q125303432}}
* &lt;!-- Anthony J. Gaughan (2022-12-30) Voting in Colonial and Revolutionary America--&gt;{{cite Q|Q125260096}}
* &lt;!-- Graves (2005) Violence, Nonviolence, and the American Revolution--&gt;{{cite Q|Q58635938}} 
* &lt;!-- Graves (2014) Restrict secrecy more than data collection--&gt;{{cite Q|Q106512569}}
* &lt;!-- Graves (2021) Congressional Gold Medals for Assange, Hale, Barlow, Winner, Manning, Edmonds, Sterling, Drake, Snowden, Ellsberg--&gt;{{cite Q|Q125570226}}
* &lt;!-- Annie Grayer (2024-03-29) &quot;Progressive Jewish lawmakers caught in the middle of bitter Democratic divide over Israel&quot;, CNN--&gt;{{cite Q|Q125247599}}
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== Notes ==
{{reflist}}
[[Category:Government]]
[[Category:News]] 
[[Category:Original research]]
[[Category:Research]]
[[Category:Political science]]
[[Category:Media]]
[[Category:Freedom and abundance]]
[[Category:Economics]]
[[Category:Political economy]]
[[Category:News]]
[[Category:Corruption]]
[[Category:Democracy]]
[[Category:war]]
[[Category:violence]]
[[Category:terrorism]]
[[Category:Conflicts]]
[[Category:nonviolence]]
[[Category:State of Palestine]]</text>
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  <page>
    <title>WikiJournal Preprints/Design and validation of an open access, 3D printed dermatoscope</title>
    <ns>0</ns>
    <id>304744</id>
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      <text bytes="25413" xml:space="preserve">{{Article info
| journal = WikiJournal of Medicine      &lt;!-- WikiJournal of Medicine, Science, or Humanities --&gt;
| last1 = Capobussi
| orcid1 = 0000-0002-1762-0471
| first1 = Matteo
| last2 = Moja
| first2 = Lorenzo
| last3 = 
| first3 = 
| last4 = 
| first4 = &lt;!-- up to 9 authors can be added in this above format --&gt;
| et_al = &lt;!-- if there are &gt;9 authors, hyperlink to the list here --&gt;
| affiliation1 = Interuniversity Center in Clinical Research, University of Milan, Via Pascal 36, 20133, Milan, Italy
| correspondence1 = matteo.capobussi@gmail.com
| affiliations = Department of Biomedical Sciences for Health, University of Milan, Via Pascal 36, 20133, Milan, Italy
| correspondence = matteo.capobussi@gmail.com
| keywords = 3D printing, Prototyping, Emergency Medicine, Dermatoscope, Maker Culture
| license = &lt;!-- default is CC-BY --&gt;
| abstract = There is a growing demand for low-cost, validated medical instruments, which are increasingly crucial in low-income settings or during health crises. These instruments can help reduce inequities and improve access to medical care. This paper provides a description of the prototyping process for a 3D-printed, open-source dermatoscope. It includes detailed building instructions and a thorough evaluation of its performance compared to a commercial solution. By utilizing this technology, general practitioners and primary care doctors can easily obtain medical equipment that is typically reserved for specialists. The dermatoscope we developed seems to be adequate for initial clinical evaluations. However, it's important to note that the overall quality of self-manufactured devices may vary depending on the available materials and the proficiency in assembly. Whenever possible, it is advisable to prioritize medical-grade certified instruments over self-made solutions.
}}

==Introduction==
[[File:Dermatoscope Prototype.jpg|left|thumb|3D model of the prototype and visual comparison with a commercial dermatoscope
(attribution: Matteo Capobussi, [https://creativecommons.org/licenses/by/3.0/deed.en CC-BY 3.0])]]
Distributed digital manufacturing of free and open-source medical hardware is gaining momentum &lt;ref&gt;A. Pavlosky, J. Glauche, S. Chambers, M. Al-Alawi, K. Yanev, T. Loubani, Validation of an effective, low cost, Free/open access 3D-printed stethoscope. PLoS ONE 13(3) (2018) e0193087. &lt;nowiki&gt;https://doi.org/10.1371/journal.pone.0193087&lt;/nowiki&gt;&lt;/ref&gt;. The increased availability of 3D printing technology, coupled with the growing number of enthusiasts and small enterprises utilizing these technologies, has led to local small-scale production of various medical devices. Some of these productions have been proposed to address crises such as war zones, disaster-stricken areas, and more recently, shortages caused by the COVID-19 pandemic &lt;ref&gt;R. Tino, R. Moore, S. Antoline, COVID-19 and the role of 3D printing in medicine. 3D Print Med 6, 11 (2020).&lt;/ref&gt; &lt;ref&gt;{{Cite journal|last=Capobussi|first=Matteo|last2=Moja|first2=Lorenzo|date=2020-11-02|title=3D printing technology and internet of things prototyping in family practice: building pulse oximeters during COVID-19 pandemic|url=https://doi.org/10.1186/s41205-020-00086-1|journal=3D Printing in Medicine|volume=6|issue=1|pages=32|doi=10.1186/s41205-020-00086-1|issn=2365-6271|pmc=PMC7605335|pmid=33136214}}&lt;/ref&gt;. In these situations, simple, low-tech, and easy-to-build medical items like stethoscopes or otoscopes can be crucial in providing adequate medical care to those in need &lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Capobussi|first=Matteo|last2=Moja|first2=Lorenzo|date=2021-11-17|title=An open-access and inexpensive 3D printed otoscope for low-resource settings and health crises|url=https://doi.org/10.1186/s41205-021-00127-3|journal=3D Printing in Medicine|volume=7|issue=1|pages=36|doi=10.1186/s41205-021-00127-3|issn=2365-6271|pmc=PMC8595962|pmid=34787772}}&lt;/ref&gt;. In developing countries, shortages of medical equipment may be more pervasive than occasional, resulting in limited access and deep inequalities.

Hobbyists and Makers worldwide possess the technology and skills to design and assemble complex and reliable prototypes for medical use. While their reverse-engineering capabilities have shown increasing success in building advanced devices such as robots or moving prosthetic arms in recent years &lt;ref&gt;{{Cite journal|last=Jones|first=G. K.|last2=Rosendo|first2=A.|last3=Stopforth|first3=R.|date=2017-07|title=Prosthetic design directives: Low-cost hands within reach|url=https://ieeexplore.ieee.org/document/8009464/|publisher=IEEE|pages=1524–1530|doi=10.1109/ICORR.2017.8009464|isbn=978-1-5386-2296-4}}&lt;/ref&gt; &lt;ref&gt;CL. Ventola, Medical Applications for 3D Printing: Current and Projected Uses. Pharm Ther. (2014) 39: 704±711.&lt;/ref&gt;, there is a need for the earlier development of validated models for various basic medical equipment pieces.

The aim of this paper is to describe the prototyping process and evaluate the performance of an open-source, 3D-printed, low-cost dermatoscope compared to a commercial solution. We specifically chose this medical device for several reasons. Its design has undergone minimal evolution, and it can be easily reproduced using commonly available electronics and magnifying systems for Makers. The cost of commercially available, medical-grade devices remains high, ranging from a few hundred to several thousand euros. This poses a barrier for general practitioners and emergency doctors, resulting in only a minority of them adopting this tool in their practice &lt;ref&gt;{{Cite journal|last=Jones|first=O.T.|last2=Jurascheck|first2=L.C.|last3=Utukuri|first3=M.|last4=Pannebakker|first4=M.M.|last5=Emery|first5=J.|last6=Walter|first6=F.M.|date=2019-09|title=Dermoscopy use in UK primary care: a survey of GP s with a special interest in dermatology|url=https://onlinelibrary.wiley.com/doi/10.1111/jdv.15614|journal=Journal of the European Academy of Dermatology and Venereology|language=en|volume=33|issue=9|pages=1706–1712|doi=10.1111/jdv.15614|issn=0926-9959|pmc=PMC6767170|pmid=30977937}}&lt;/ref&gt;. By openly publishing this prototype, we hope to enable others to replicate and potentially improve it.


== Material and methods ==

Our dermatoscope was designed with the goal of maintaining quality standards similar to current professional devices while significantly reducing production costs. The entire design was created using a free online CAD tool called TinkerCAD (www.tinkercad.com, Autodesk Inc., San Rafael, California, USA), and it was manufactured using a Prusa i3 pro B FDM 3D printer (Geeetech Ltd, Shenzhen, China) and an SLA Mars Pro (Elegoo Inc., Shenzhen, China). Detailed building instructions can be found in Appendix A.

A dermatoscope consists of two main parts: a head that houses the lighting and magnification systems, and a battery compartment for handling the device. We made the decision to design the head and handle separately. For the head, we utilized the same battery compartment as a low-cost 3D-printed otoscope that we had previously developed &lt;ref name=&quot;:0&quot; /&gt;. This allows for multiple heads to be used with the same handle, easily transitioning from an otoscope to a dermatoscope. In case of irreparable damage, the heads can be replaced. The development of a shared platform for multiple devices could lead to a reduction in overall production costs, a concept also employed in the car industry &lt;ref&gt;{{Cite journal|date=2024-03-19|title=Volkswagen Group MQB platform|url=https://en.wikipedia.org/w/index.php?title=Volkswagen_Group_MQB_platform&amp;oldid=1214538734|journal=Wikipedia|language=en}}&lt;/ref&gt;.

[[File:Dermatoscope prototype resolution test a.jpg|left|thumb|220x220px|Comparison using a photographic lens quality scale (left: 6x prototype; right: 10x medical device)
(attribution: Matteo Capobussi, [[creativecommons:by/3.0/deed.en|CC-BY 3.0]])]]
To facilitate component retrieval, the power for the dermatoscope is supplied by two common AA batteries, connected using ball pen springs and a layer of tin. 

For the lighting system, we chose to use 8 white LEDs, which are typically already available in a Maker's toolbox. To ensure water resistance, the LEDs need to be placed in a transparent enclosure. While FDM clear filaments are an option, we opted for stereolithography resin to guarantee waterproofing of the head. 

In our design, we incorporated an adjustable optical block consisting of plastic Fresnel lenses, which are affordable and easily obtainable from e-commerce websites. To facilitate skin contact and enable the epiluminescence effect after applying oil, we added a plexiglass layer. The internal face of the plexiglass features an engraved millimeter scale for easy measurement of detected skin lesions. By utilizing these components, we were able to keep the costs around 5 euros (Table 1). 

[[File:Dermatoscope prototype resolution test b.jpg|left|thumb|221x221px|Comparison using a photographic lens quality scale (left: 6x prototype; right: 10x medical device) (attribution: Matteo Capobussi, [[creativecommons:by/3.0/deed.en|CC-BY 3.0]])]]
{| class=&quot;wikitable mw-collapsible&quot;
|+
|'''Component''' 
|'''Number'''
|'''Cost per  unit -currency'''
|'''Total cost  -'''

'''currency'''
|'''Source of  materials'''
|'''Material  type'''
|-
|'''''Handle'''''
|
|
|
|
|
|-
|&lt;nowiki&gt;- 3D printed parts: SLA (FDM)&lt;/nowiki&gt;
|/
|/
|1.28 (0.64) €
|/
|Polymer
|-
|&lt;nowiki&gt;- SLA (FDM) printing electricity  consumption &lt;/nowiki&gt;
|/
|/
|0.07 (0.12) €
|/
|Other
|-
|&lt;nowiki&gt;- nickel strip&lt;/nowiki&gt;
|7cm
|0.006 € /  cm
|0.04 €
|&lt;nowiki&gt;https://www.amazon.it/gp/product/B07MM88TJZ/&lt;/nowiki&gt;
|Metal
|-
|&lt;nowiki&gt;- 3-pin  switch&lt;/nowiki&gt;
|1
|/
|0.09 €
|&lt;nowiki&gt;https://www.amazon.it/gp/product/B06XGTKK9Z/&lt;/nowiki&gt;
|Other
|-
|&lt;nowiki&gt;- 28  AWG Electric wiring&lt;/nowiki&gt;
|16cm
|0.002 € /  cm
|0.03 €
|&lt;nowiki&gt;https://www.amazon.it/gp/product/B07PN8GD6G/&lt;/nowiki&gt;
|Other
|-
|&lt;nowiki&gt;- M3 15 mm screws  (side of the handle)&lt;/nowiki&gt;
|2 
|0.01 €
|0.03 €
|/
|Metal
|-
|&lt;nowiki&gt;- M2  6 mm screws (switch and battery compartment)&lt;/nowiki&gt;
|3
|0.01 €
|0.03 €
|/
|Metal
|-
|&lt;nowiki&gt;- Recycled spring  from ball-pen&lt;/nowiki&gt;
|2
|/
|0 €
|/
|Metal
|-
|'''Total SLA (FDM)'''
|/
|/
|'''1.57 (0.98)''' €
|/
|/
|-
|
|
|
|
|
|
|-
|'''''Dermatoscope Head'''''
|
|
|
|
|
|-
|&lt;nowiki&gt;- 3D printed parts: SLA (FDM)&lt;/nowiki&gt;
|/
|/
|1.61 (0.80) €
|/
|Polymer
|-
|&lt;nowiki&gt;- SLA (FDM) printing electricity  consumption&lt;/nowiki&gt;
|/
|/
|0.05 (0.09) €
|/
|Other
|-
|&lt;nowiki&gt;- white  5mm, 3.3V LEDs&lt;/nowiki&gt;
|8
|0.05 €
|0.4 €
|/
|Other
|-
|&lt;nowiki&gt;- 28 AWG Electric wiring&lt;/nowiki&gt;
|12 cm
|0.002 € /  cm
|0.02 €
|&lt;nowiki&gt;https://www.amazon.it/gp/product/B07PN8GD6G/&lt;/nowiki&gt;
|Other
|-
|&lt;nowiki&gt;- nickel strip&lt;/nowiki&gt;
|3 cm
|0.006 € /  cm
|0.02 €
|
|Metal
|-
|&lt;nowiki&gt;- Fresnel lens,  magnification factor 3x, 1mm thickness&lt;/nowiki&gt;
|2
|0.95 €
|1.9 €
|&lt;nowiki&gt;https://www.amazon.it/Ingrandimento-Formato-Magnifier-Raccoglitore-Firestarter/dp/B06W5FCS4Q&lt;/nowiki&gt;
|Polymer
|-
|&lt;nowiki&gt;- Recycled plexiglass/PVC  layer,  33x33mm (i.e. from a plastic  bottle)&lt;/nowiki&gt;
|1
|/
|/
|
|Polymer
|-
|'''Total SLA (FDM)'''
|/
|/
|'''4.00 (3.23)''' €
|/
|/
|}


An alternative to Fresnel lenses is to use a printable optical lens system. Theoretically, one could take advantage of the high-resolution stereolithography printers provide to produce the optical block of an entire photographic camera &lt;ref&gt;{{Cite web|url=https://formlabs.com/blog/creating-camera-lenses-with-stereolithography/|title=Creating 3D Printed Lenses and a 3D Printed Camera with Stereolithography|website=Formlabs|language=en|access-date=2024-04-16}}&lt;/ref&gt;. However, the post-printing process required to achieve sufficient visual quality is complex and time-consuming, and it relies on specific and expensive optical resin. Using stereolithography with standard (non-optical quality) clear resin to manufacture the entire device increases the overall cost, as the resin is sold at around 30€/L. On the other hand, for fused deposition modeling, 1 kg of ABS filament can be purchased online for around 15€. The estimated average power consumption is 0.05 kWh for FDM printing and 0.03 kWh for stereolithography &lt;ref&gt;S. Walls, J. Corney, G. Vasantha, Relative energy consumption of low-cost 3d printers. Proceedings of the 12th International Conference on Manufacturing Research (2014) &lt;/ref&gt;.

== Results ==

Medical literature does not provide standardized parameters for comparing the performance of dermatoscopes. Typically, these devices are described in terms of technical features such as magnifying power and light intensity. In our study, we compared our low-cost prototype to an entry-level commercial solution (dermatoscope Gima 2000, G.I.M.A. S.p.A., Milan, Italy) using these parameters, as well as evaluating the field of view and color quality.

The 3D printed dermatoscope functions in the same way as traditional dermatoscopes. Healthcare professionals simply need to turn on the light using the switch and gently but firmly press the dermatoscope head against the patient's skin. When the epiluminescence effect is desired, oil should be applied since the plexiglass interface is not polarized. Adjusting the helix provides focus adjustment when needed, and removing the lens holder allows for changing the magnification level from 6x to 3x.

Our analysis revealed some differences between the prototype and the certified dermatoscope. Fresnel lenses can only provide limited magnification power without distorting the image. The prototype was able to achieve 6x magnification, while the glass lenses of the commercial dermatoscope provided 10x magnification without chromatic aberrations. However, it's important to note that magnification alone does not guarantee clarity of vision. For this reason, we tested both devices using an imaging-quality millimeter scale designed for comparing photographic lenses &lt;ref&gt;G. Ferzetti, Conoscere le Leica. Gieffe edizioni, Rovigo, 1984&lt;/ref&gt;. Both our prototype and the Gima dermatoscope were able to clearly visualize the gaps between the smallest test lines, which were placed at a 0.2mm distance. This, combined with a comparable field of view (prototype: 25 mm; medical device: 26mm), resulted in a smaller but clear image that was sufficient for evaluating the details of most skin lesions, which are typically at least 10 times larger than the measured minimum resolution.

The total amount of light emitted by the instruments was measured using a professional exposimeter (Bowens flash meter III, Sekonic Electronics Inc, Japan). Our prototype emitted 5000 lux, significantly higher than the Gima dermatoscope, which only produced 1400 lux. The commercial dermatoscope was equipped with a dimmer to adjust the amount of light according to the characteristics of the detected skin lesions.

There was also a difference in color temperature. Our white LEDs emitted a cool light (7600K), whereas the halogen lamp of the certified dermatoscope consistently emitted a warmer light (4700K). Although LED dermatoscopes have become the industry standard, it remains unclear whether the disparities in color temperature can interfere with diagnosis. It is a common experience for doctors to have to adapt to the technical specifications of a new instrument, and to address this issue, attempts have been made to standardize imaging &lt;ref&gt;{{Cite journal|last=Badano|first=Aldo|last2=Revie|first2=Craig|last3=Casertano|first3=Andrew|last4=Cheng|first4=Wei-Chung|last5=Green|first5=Phil|last6=Kimpe|first6=Tom|last7=Krupinski|first7=Elizabeth|last8=Sisson|first8=Christye|last9=Skrøvseth|first9=Stein|date=2015-02-01|title=Consistency and Standardization of Color in Medical Imaging: a Consensus Report|url=https://doi.org/10.1007/s10278-014-9721-0|journal=Journal of Digital Imaging|language=en|volume=28|issue=1|pages=41–52|doi=10.1007/s10278-014-9721-0|issn=1618-727X|pmc=PMC4305059|pmid=25005868}}&lt;/ref&gt;.

[[File:Dermatoscope performance comparison G channel a.png|left|thumb|221x221px|Comparison between devices and Bland-Altman plots for the “green” color channel
(attribution: Matteo Capobussi, [[creativecommons:by/3.0/deed.en|CC-BY 3.0]])]]
[[File:Dermatoscope performance comparison G channel b.png|left|thumb|221x221px|Comparison between devices and Bland-Altman plots for the “green” color channel (attribution: Matteo Capobussi, [[creativecommons:by/3.0/deed.en|CC-BY 3.0]])]]
[[File:Dermatoscope performance comparison G channel c.png|left|thumb|221x221px|Comparison between devices and Bland-Altman plots for the “green” color channel (attribution: Matteo Capobussi, [[creativecommons:by/3.0/deed.en|CC-BY 3.0]])]]

=== '''Color comparison''' ===
To compare the performance of the two devices on skin lesions of different colors, we utilized a color scale chart - specifically von Luschan's skin chromatic scale - which has a high correlation with skin color evaluation conducted by a reflectance spectrophotometer &lt;ref&gt;A. Treesirichod, S. Chansakulporn, P. Wattanapan, Correlation between skin color evaluation by skin color scale chart and narrowband reflectance spectrophotometer. Indian J Dermatol (2014) 59:339-42&lt;/ref&gt;. We printed a copy of the scale and captured digital images of its 36 items using both devices at their brightest lamp setting. Subsequently, the images were analyzed using Photoshop CS6 (Adobe Inc., San Jose, California, USA), which provided a 3-point RGB component profile for each image, enabling us to examine each color individually. We compared these data with the color profiles obtained directly from the chart and computed the Pearson's correlation coefficient for each color and device. The results demonstrated a strong correlation for every comparison, with the prototype exhibiting a higher correlation than the Gima dermatoscope. For example, in the &quot;green&quot; channel, the correlation coefficient (r) was 0.87 when comparing the certified medical device to the chart and 0.97 when using our prototype. Similar patterns were observed for the other channels: the &quot;red&quot; correlation coefficients were 0.73 and 0.96, and the &quot;blue&quot; coefficients were 0.80 and 0.89, favoring the prototype. It is important to note that correlation should not be interpreted as agreement.

To better visualize the differences between the two measurement instruments, we employed a Bland-Altman plot &lt;ref&gt;DG. Altman, JM. Bland, Measurement in medicine: the analysis of method comparison studies. Statistician (1983) 32:307–17. 10.2307/2987937&lt;/ref&gt;. This graphical tool is typically used to compare laboratory diagnostic tests, plotting the means of each pair of measurements on the x-axis against the percentage of differences between the measurements on the y-axis. In all the graphs, an increase in variability of the differences was observed as the magnitude of the measurement increased. This was likely caused by an elevated level of reflections produced by the brighter items on von Luschan's scale. The trend line consistently showed a positive slope: for lower (darker) values, both devices produced brighter colors compared to the chart values, while for the brightest values, the lines converged towards zero, indicating a more accurate color representation. The majority of measurements fell within the confidence intervals, suggesting that both devices yielded accurate results. However, this was not the case for the &quot;red&quot; channel. The data from this component did not follow a normal distribution, which is a requirement for the Bland-Altman plot. We identified the cause as the brightness setting chosen for the measurements, as depicted in the graphs where many values reached 255, the maximum value. To overcome this issue, we plotted the Bland-Altman graph using the 2.5 and 97.5 percentiles &lt;ref&gt;ME. Frey, HC. Petersen, O. Gerke, Nonparametric Limits of Agreement for Small to Moderate Sample Sizes: a Simulation Study. Stats (2020) 3, 343–355, 10.3390/stats3030022&lt;/ref&gt;. These graphs closely resembled those of other colors, with linearly increasing measurements.

Overall, we can confidently state that both devices provided an accurate representation of the colors included in the skin chart, albeit with some differences related to the color temperature of the lighting system used.


== Discussion ==

By utilizing 3D printing and leveraging Makers’ prototyping expertise, we successfully designed and fabricated a functional dermatoscope at a cost of approximately 5€. Currently, branded dermatoscopes are expensive, which can pose a barrier for clinicians, particularly in non-specialized contexts. Given the well-established working principles of this device, it was relatively easy to implement the design using modern electronics. The cost of our instrument was 40 to 80 times lower compared to professional options. Our project implemented a modular platform, enabling interchangeability between different specialized heads, further reducing costs for small-scale production. While the performance of our dermatoscope was similar to entry-level commercial solutions, it did have some limitations. The magnification power was lower compared to glass optics, as the Fresnel lenses only achieved a 6x image enlargement instead of the professional solution's 10x. However, our dermatoscope was capable of detecting details as small as 0.2mm, making it suitable for most skin lesions. Color profiles were similar between devices, but some differences emerged at extreme brightness levels, deviating from the reference table. This can be addressed by incorporating a dimmer, which is already present in the certified device. In our case, Makers could easily refine our design by, for example, adding a potentiometer.

To ensure safety, every possible measure to prevent harm to patients should be taken. For instance, the materials used should be washable and antibacterial. FDM 3D prints are not waterproof or sterilizable, and stereolithography resin is toxic in its liquid form. However, the dermatoscope only makes contact with the patient's skin through the central part of its head, where an easily cleanable and non-toxic glass/plexiglass square is positioned. When electronics are involved, the final product should be waterproof to prevent potential short circuits. Only a harmless, low voltage (3V) current is used. Therefore, we can assume that the safety criteria are met.

When considering emergency contexts, dermoscopy may not immediately come to mind. So far, the focus of developing 3D printed instruments has been on items such as stethoscopes and prosthetic arms. However, one of the lessons learned from the COVID-19 pandemic is that ordinary and chronic diseases may be neglected throughout the duration of an emergency, compromising patient management and increasing mortality rates &lt;ref&gt;{{Cite journal|last=Huet|first=Fabien|last2=Prieur|first2=Cyril|last3=Schurtz|first3=Guillaume|last4=Gerbaud|first4=Edouard|last5=Manzo-Silberman|first5=Stéphane|last6=Vanzetto|first6=Gerald|last7=Elbaz|first7=Meyer|last8=Tea|first8=Victoria|last9=Mercier|first9=Grégoire|date=2020-05|title=One train may hide another: Acute cardiovascular diseases could be neglected because of the COVID-19 pandemic|url=https://linkinghub.elsevier.com/retrieve/pii/S1875213620300991|journal=Archives of Cardiovascular Diseases|language=en|volume=113|issue=5|pages=303–307|doi=10.1016/j.acvd.2020.04.002|pmc=PMC7186196|pmid=32362433}}&lt;/ref&gt;. Increased availability of these instruments could prove beneficial in situations where the duration of the emergency is unknown.

== Conclusions ==

The proliferation of 3D printed instruments has the potential to reduce inequalities by providing more accessible medical diagnostic tools. With this technology, general practitioners and primary care doctors can readily obtain medical equipment that is typically reserved for specialists. While general practitioners may not possess the specialized training required for complex diagnoses, they can use these instruments as a preliminary assessment tool, enabling them to differentiate between common diseases and those that require specialist attention. Without such instruments, primary care doctors might unnecessarily seek specialist consultations, leading to increased costs for patients and healthcare systems. Further research should focus on designing and validating additional equipment. The dermatoscope we have developed appears to be adequate for initial clinical evaluations. However, the overall quality of self-manufactured devices may vary depending on the available materials and assembly proficiency. Whenever possible, it is advisable to prioritize medical-grade certified instruments over self-made solutions.

==Additional information==

===Competing interests===
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

===Ethics statement===
No human or animal subjects were involved in this research.

==References==
All design files are published in Mendeley repository: Capobussi, Matteo (2023), “3D printed dermatoscope stl files”, Mendeley Data, V1, doi: 10.17632/6ttjr5vx94.1</text>
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  <page>
    <title>Talk:Net neutrality and 'Restoring Internet freedom'</title>
    <ns>1</ns>
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      <id>2622870</id>
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      <timestamp>2024-04-25T13:30:44Z</timestamp>
      <contributor>
        <username>DavidMCEddy</username>
        <id>218607</id>
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      <model>wikitext</model>
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      <text bytes="4616" xml:space="preserve">{{ping|DavidMCEddy}} &quot;''After being relatively stable for the 50 years from 1925 to 1975, the incarceration rate in the US shot up by a factor of five in the last quarter of the twentieth century. This increase in incarcerations occurred without a corresponding change in crime rates. This change has been explained as a product of decisions by mainstream commercial broadcasters to focus on the police blotter while firing nearly all their investigative journalists. A few popular programs like “60 Minutes” were exceptions.[14]''&quot; That's interesting, though it's not exactly clear why an increase in crime reporting would cause an increase in incarcerations. The Wikipedia [https://en.wikipedia.org/wiki/United_States_incarceration_rate#Editorial_policies_of_major_media article] states that high crime reporting is associated with increased sentence length, but that's not very specific. What is the causal relationship here? [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 19:17, 23 April 2024 (UTC)

:Please excuse: To me, it's fairly clear. It might be better stated in the Wikiversity article on [[Confirmation bias and conflict]]: Walter Lippmann in his 1922 book on ''[[w:Public Opinion (book)|Public Opinion]]'' said, &quot;The real environment is altogether too big, too complex, and too fleeting for direct acquaintance&quot; between people and their environment. Each person constructs a pseudo-environment that is a subjective, biased, and necessarily abridged mental image of the world that is unique to that individual and evolves over time. People &quot;live [and act] in the same world, but they think and feel [and decide] in different ones.&quot; The increase in incarceration rate visible in that plot seems to have been a product of the [[Confirmation bias and conflict#Corruption trilogy|corruption trilogy]], mentioned in that article. 

:Does this make sense? Thanks, [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 01:01, 24 April 2024 (UTC)
::Not really. Do you mean that crime reporting increases crime itself by power of suggestion, the public's tendency to report/convict others of crime, or by some other mechanism? [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 04:54, 24 April 2024 (UTC)

:::I just added the following: 
:::* ''The incarceration rate is a function of public perception of crime, which is unrelated to the crime rate,'' at least in the US between 1925 and 2014.
:::How's this? 
:::[[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 13:18, 24 April 2024 (UTC)
::::That doesn't answer my question. I am asking ''how'' public perception of crime, or rather, crime reporting, affects incarceration rates. By what means? Your sentence also is not strictly true, as incarceration rate and public perception of crime cannot be entirely independent of actual crime. [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 11:44, 25 April 2024 (UTC)
:::: To be clear, I'm not saying that crime reporting does not affect incarceration rates. I'm just curious how. This is not obvious to me. To rephrase the question once more, how does crime reporting increase incarceration rate? I can speculate, and certainly it does not seem an implausible claim, but presumably the authors of these studies have suggested why this is the case. [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 11:48, 25 April 2024 (UTC)

::::: Thanks. I added another paragraph plus a few references. I think it can be documented that the crime rate would fall dramatically if we had 100% public financing for quality child care from pregnancy through age 17. In addition, the entire population would be healthier, per &lt;!-- Inequality Kills Us All: COVID-19's Health Lessons for the World--&gt;{{cite Q|Q118236554}}, and after a few decades, the entire education system would become free, paid by productivity increases we would not have without it, according to substantial research by [[w:Eric Hanushek|Eric Hanushek]] and co-workers. However, none of that incarcerations, unless we actually changed the structure of the media. For moore on the media, see [[Information is a public good: Designing experiments to improve government]]. Hope this helps. [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 13:30, 25 April 2024 (UTC)</text>
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      <contributor>
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      <comment>Reply</comment>
      <model>wikitext</model>
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      <text bytes="4749" xml:space="preserve">{{ping|DavidMCEddy}} &quot;''After being relatively stable for the 50 years from 1925 to 1975, the incarceration rate in the US shot up by a factor of five in the last quarter of the twentieth century. This increase in incarcerations occurred without a corresponding change in crime rates. This change has been explained as a product of decisions by mainstream commercial broadcasters to focus on the police blotter while firing nearly all their investigative journalists. A few popular programs like “60 Minutes” were exceptions.[14]''&quot; That's interesting, though it's not exactly clear why an increase in crime reporting would cause an increase in incarcerations. The Wikipedia [https://en.wikipedia.org/wiki/United_States_incarceration_rate#Editorial_policies_of_major_media article] states that high crime reporting is associated with increased sentence length, but that's not very specific. What is the causal relationship here? [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 19:17, 23 April 2024 (UTC)

:Please excuse: To me, it's fairly clear. It might be better stated in the Wikiversity article on [[Confirmation bias and conflict]]: Walter Lippmann in his 1922 book on ''[[w:Public Opinion (book)|Public Opinion]]'' said, &quot;The real environment is altogether too big, too complex, and too fleeting for direct acquaintance&quot; between people and their environment. Each person constructs a pseudo-environment that is a subjective, biased, and necessarily abridged mental image of the world that is unique to that individual and evolves over time. People &quot;live [and act] in the same world, but they think and feel [and decide] in different ones.&quot; The increase in incarceration rate visible in that plot seems to have been a product of the [[Confirmation bias and conflict#Corruption trilogy|corruption trilogy]], mentioned in that article. 

:Does this make sense? Thanks, [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 01:01, 24 April 2024 (UTC)
::Not really. Do you mean that crime reporting increases crime itself by power of suggestion, the public's tendency to report/convict others of crime, or by some other mechanism? [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 04:54, 24 April 2024 (UTC)

:::I just added the following: 
:::* ''The incarceration rate is a function of public perception of crime, which is unrelated to the crime rate,'' at least in the US between 1925 and 2014.
:::How's this? 
:::[[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 13:18, 24 April 2024 (UTC)
::::That doesn't answer my question. I am asking ''how'' public perception of crime, or rather, crime reporting, affects incarceration rates. By what means? Your sentence also is not strictly true, as incarceration rate and public perception of crime cannot be entirely independent of actual crime. [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 11:44, 25 April 2024 (UTC)
:::: To be clear, I'm not saying that crime reporting does not affect incarceration rates. I'm just curious how. This is not obvious to me. To rephrase the question once more, how does crime reporting increase incarceration rate? I can speculate, and certainly it does not seem an implausible claim, but presumably the authors of these studies have suggested why this is the case. [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 11:48, 25 April 2024 (UTC)

::::: Thanks. I added another paragraph plus a few references. I think it can be documented that the crime rate would fall dramatically if we had 100% public financing for quality child care from pregnancy through age 17. In addition, the entire population would be healthier, per &lt;!-- Inequality Kills Us All: COVID-19's Health Lessons for the World--&gt;{{cite Q|Q118236554}}, and after a few decades, the entire education system would become free, paid by productivity increases we would not have without it, according to substantial research by [[w:Eric Hanushek|Eric Hanushek]] and co-workers. However, none of that incarcerations, unless we actually changed the structure of the media. For moore on the media, see [[Information is a public good: Designing experiments to improve government]]. Hope this helps. [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 13:30, 25 April 2024 (UTC)
::::::??? [[User:AP295|AP295]] ([[User talk:AP295|discuss]] • [[Special:Contributions/AP295|contribs]]) 13:38, 25 April 2024 (UTC)</text>
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    <title>File:Link.5.LibraryRPATH.20240425-2.pdf</title>
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      <text bytes="282" xml:space="preserve">== Summary ==
{{Information
|Description=Link.5: Library Search Using -rpath (20240425-2 - 20240425)
|Source={{own|Young1lim}}
|Date=2024-04-25
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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  <page>
    <title>User:Addemf/sandbox/Technical Reasoning/Naive Set Theory 2</title>
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      <comment>/* Intersection */</comment>
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      <text bytes="2257" xml:space="preserve">== Set Operations ==

=== Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
=== Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements ''which are in both''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}</text>
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    <revision>
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      <comment>/* Set Operations */</comment>
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      <text bytes="3998" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}</text>
      <sha1>fanr2kgdqw0udureqe7ir31k7cst09o</sha1>
    </revision>
    <revision>
      <id>2622881</id>
      <parentid>2622880</parentid>
      <timestamp>2024-04-25T16:27:04Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Set-Difference */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4439" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', find 
: &lt;math&gt;(A\smallsetminus B)\cap (B\smallsetminus A)&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', consider the equations
: &lt;math&gt; A\cup B = (A\smallsetminus B)\cup B&lt;/math&gt;
: &lt;math&gt;A\cup B = (A\smallsetminus B)\cup (B\smallsetminus A)&lt;/math&gt;

For each equation, is it always true?

Is it sometimes true?
{{robelbox/close}}</text>
      <sha1>nb7zqx6tf6xvkh6lnj9jlyppeta4x8t</sha1>
    </revision>
    <revision>
      <id>2622882</id>
      <parentid>2622881</parentid>
      <timestamp>2024-04-25T16:35:12Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Pairwise Union */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4817" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Can we solve set equations the way that we solve algebraic equations?

Suppose that ''X'' is some set which satisfies the equation 
: &lt;math&gt;X \cup \{1,2,3\}=\{-2,1,2,3,4\}&lt;/math&gt;

Can we determine what ''X'' must be?

If we cannot infer what ''X'' is, then can we at least make some inference about the elements of ''X''?
{{robelbox/close}}

=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', find 
: &lt;math&gt;(A\smallsetminus B)\cap (B\smallsetminus A)&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', consider the equations
: &lt;math&gt; A\cup B = (A\smallsetminus B)\cup B&lt;/math&gt;
: &lt;math&gt;A\cup B = (A\smallsetminus B)\cup (B\smallsetminus A)&lt;/math&gt;

For each equation, is it always true?

Is it sometimes true?
{{robelbox/close}}</text>
      <sha1>csxcdxdz5idjmkh125z816v0foxodpz</sha1>
    </revision>
    <revision>
      <id>2622891</id>
      <parentid>2622882</parentid>
      <timestamp>2024-04-25T17:56:39Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7622" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Can we solve set equations the way that we solve algebraic equations?

Suppose that ''X'' is some set which satisfies the equation 
: &lt;math&gt;X \cup \{1,2,3\}=\{-2,1,2,3,4\}&lt;/math&gt;

Can we determine what ''X'' must be?

If we cannot infer what ''X'' is, then can we at least make some inference about the elements of ''X''?
{{robelbox/close}}

=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', find 
: &lt;math&gt;(A\smallsetminus B)\cap (B\smallsetminus A)&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', consider the equations
: &lt;math&gt; A\cup B = (A\smallsetminus B)\cup B&lt;/math&gt;
: &lt;math&gt;A\cup B = (A\smallsetminus B)\cup (B\smallsetminus A)&lt;/math&gt;

For each equation, is it always true?

Is it sometimes true?
{{robelbox/close}}
=== Universal Set and Complement ===
In many settings, we will be exclusively interested in sets which are all subsets of a single set.  

For instance, in some settings we will only be interested in the natural numbers, and all of the sets that we consider will be subsets of it.

Or we might only be interested in the set of real numbers, or other kinds of sets, like the set of all matrices.

Whenever we restrict our interests to the subsets of one particular set, we call that one set the &quot;universal set&quot;.  We may choose the universal set to be any set that we like.

Once we specify a universal set, we can then define the notion of &quot;the complement of a set&quot;.

Suppose that we choose the universal set &lt;math&gt;U=\{0,1,2,3,4\}&lt;/math&gt; and consider the subset &lt;math&gt;A=\{2,3\}&lt;/math&gt;.  Then the complement of ''A'' is the set of all elements in the universe which are ''not'' in ''A''.
: &lt;math&gt;A^c = \{0,1,4\}&lt;/math&gt;

Notice that the choice of universal set will affect the complement of a given set.  If the universal set is &lt;math&gt;U=\{1,2,3,4\}&lt;/math&gt; and &lt;math&gt;A=\{2,3\}&lt;/math&gt; then 
: &lt;math&gt;A^c = \{1,4\}&lt;/math&gt;
{{robelbox|title=Exercise|theme=2}}
Show that for any universal set ''U'' and subset &lt;math&gt;A\subseteq U&lt;/math&gt;, 
: &lt;math&gt;A^c = U\smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the universal set &lt;math&gt;U=\Bbb N&lt;/math&gt; and subsets &lt;math&gt;A=\{2,4,6,\dots\}&lt;/math&gt; and &lt;math&gt;B = \{1,2\}&lt;/math&gt;.

Find the following sets.

(1.) &lt;math&gt;A^c&lt;/math&gt;

(2.) &lt;math&gt;B^c&lt;/math&gt;

(3.) &lt;math&gt;A^c \cap B^c&lt;/math&gt;

(4.) &lt;math&gt;(A\cup B)^c&lt;/math&gt;
{{robelbox/close}}
We can now state an important relationship between the union, intersection, and complement.  
{{robelbox|title=Theorem: U-C De Morgan's}}
''Theorem'': Suppose the universal set is ''U'' with subsets ''A'' and ''B''.  Then 
: &lt;math&gt;(A\cup B)^c = A^c \cap B^c&lt;/math&gt;
----
In order to prove that two sets are equal, we must prove that every element in the left set is in the right set; and every element in the right set is in the left set.

In short, the fundamental way to prove a set equality, &lt;math&gt;X=Y&lt;/math&gt;, is to prove two subset relations.

: &lt;math&gt;X\subseteq Y&lt;/math&gt;
and 
: &lt;math&gt;Y\subseteq X&lt;/math&gt;
We now do just that, with ''X'' identified with the left side, &lt;math&gt;(A\cup B)^c&lt;/math&gt;.  ''Y'' is identified with the right side, &lt;math&gt;A^c\cap B^c&lt;/math&gt;.

''Proof'':

(For &lt;math&gt;(A\cup B)^c\subseteq A^c \cap B^c&lt;/math&gt;.)

{{small|(1.)}} Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.  

(The goal then is to show &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.)
----
{{small|(2.)}} So &lt;math&gt; x\notin A\cup B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (2.) and the definition of union.
----


{{robelbox/close}}</text>
      <sha1>2bv1y62gs3mxwv5lpd47239h7gnuzgc</sha1>
    </revision>
    <revision>
      <id>2622906</id>
      <parentid>2622891</parentid>
      <timestamp>2024-04-25T20:15:44Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Universal Set and Complement */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8912" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Can we solve set equations the way that we solve algebraic equations?

Suppose that ''X'' is some set which satisfies the equation 
: &lt;math&gt;X \cup \{1,2,3\}=\{-2,1,2,3,4\}&lt;/math&gt;

Can we determine what ''X'' must be?

If we cannot infer what ''X'' is, then can we at least make some inference about the elements of ''X''?
{{robelbox/close}}

=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', find 
: &lt;math&gt;(A\smallsetminus B)\cap (B\smallsetminus A)&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', consider the equations
: &lt;math&gt; A\cup B = (A\smallsetminus B)\cup B&lt;/math&gt;
: &lt;math&gt;A\cup B = (A\smallsetminus B)\cup (B\smallsetminus A)&lt;/math&gt;

For each equation, is it always true?

Is it sometimes true?
{{robelbox/close}}
=== Universal Set and Complement ===
In many settings, we will be exclusively interested in sets which are all subsets of a single set.  

For instance, in some settings we will only be interested in the natural numbers, and all of the sets that we consider will be subsets of it.

Or we might only be interested in the set of real numbers, or other kinds of sets, like the set of all matrices.

Whenever we restrict our interests to the subsets of one particular set, we call that one set the &quot;universal set&quot;.  We may choose the universal set to be any set that we like.

Once we specify a universal set, we can then define the notion of &quot;the complement of a set&quot;.

Suppose that we choose the universal set &lt;math&gt;U=\{0,1,2,3,4\}&lt;/math&gt; and consider the subset &lt;math&gt;A=\{2,3\}&lt;/math&gt;.  Then the complement of ''A'' is the set of all elements in the universe which are ''not'' in ''A''.
: &lt;math&gt;A^c = \{0,1,4\}&lt;/math&gt;

Notice that the choice of universal set will affect the complement of a given set.  If the universal set is &lt;math&gt;U=\{1,2,3,4\}&lt;/math&gt; and &lt;math&gt;A=\{2,3\}&lt;/math&gt; then 
: &lt;math&gt;A^c = \{1,4\}&lt;/math&gt;
{{robelbox|title=Exercise|theme=2}}
Show that for any universal set ''U'' and subset &lt;math&gt;A\subseteq U&lt;/math&gt;, 
: &lt;math&gt;A^c = U\smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the universal set &lt;math&gt;U=\Bbb N&lt;/math&gt; and subsets &lt;math&gt;A=\{2,4,6,\dots\}&lt;/math&gt; and &lt;math&gt;B = \{1,2\}&lt;/math&gt;.

Find the following sets.

(1.) &lt;math&gt;A^c&lt;/math&gt;

(2.) &lt;math&gt;B^c&lt;/math&gt;

(3.) &lt;math&gt;A^c \cap B^c&lt;/math&gt;

(4.) &lt;math&gt;(A\cup B)^c&lt;/math&gt;
{{robelbox/close}}
We can now state an important relationship between the union, intersection, and complement.  
{{robelbox|title=Theorem: U-C De Morgan's}}
''Theorem'': Suppose the universal set is ''U'' with subsets ''A'' and ''B''.  Then 
: &lt;math&gt;(A\cup B)^c = A^c \cap B^c&lt;/math&gt;
----
In order to prove that two sets are equal, we must prove that every element in the left set is in the right set; and every element in the right set is in the left set.

In short, the fundamental way to prove a set equality, &lt;math&gt;X=Y&lt;/math&gt;, is to prove two subset relations.

: &lt;math&gt;X\subseteq Y&lt;/math&gt;
and 
: &lt;math&gt;Y\subseteq X&lt;/math&gt;
We now do just that, with ''X'' identified with the left side, &lt;math&gt;(A\cup B)^c&lt;/math&gt;.  ''Y'' is identified with the right side, &lt;math&gt;A^c\cap B^c&lt;/math&gt;.

''Proof'':

(For &lt;math&gt;(A\cup B)^c\subseteq A^c \cap B^c&lt;/math&gt;.)

{{small|(1.)}} Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.  

(The goal then is to show &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.)
----
{{small|(2.)}} So &lt;math&gt; x\notin A\cup B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (2.) and the definition of union.
----
{{small|(4.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (3.) and the definition of complement.
----
{{small|(5.)}} So &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

''Reason'': From (4.) and the definition of intersection.  
----
{{small|(6.)}} So &lt;math&gt;(A\cup B)^c\subseteq A^c\cap B^c&lt;/math&gt;.

''Reason'': Lines (1.) to (5.) and the definition of subsets.
----
{{small|(7.)}} (For &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.)

Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

(The goal then is to show &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.
----
{{small|(8.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (7.) and the definition of intersection.
----
{{small|(9.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (8.) and the definition of the complement.
----
{{small|(10.)}} So &lt;math&gt;x\notin A\cup B&lt;/math&gt;.

''Reason'': From (9.) and the definition of union.
----
{{small|(11.)}} So &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.

''Reason'': From (10.) and the definition of complement.
----
{{small|(12.)}} So &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.

''Reason'': From (7.) to (11.) and the definition of subsets.
----
{{small|(13.)}} So &lt;math&gt;(A\cup B)^c = A^c\cap B^c&lt;/math&gt;.

''Reason'': From (6.) and (12.) and the definition of set equality.

{{robelbox/close}}</text>
      <sha1>j6k9rzchfe56x1xlem7vieyypxzhi58</sha1>
    </revision>
    <revision>
      <id>2622948</id>
      <parentid>2622906</parentid>
      <timestamp>2024-04-26T02:51:44Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10000" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Can we solve set equations the way that we solve algebraic equations?

Suppose that ''X'' is some set which satisfies the equation 
: &lt;math&gt;X \cup \{1,2,3\}=\{-2,1,2,3,4\}&lt;/math&gt;

Can we determine what ''X'' must be?

If we cannot infer what ''X'' is, then can we at least make some inference about the elements of ''X''?
{{robelbox/close}}

=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', find 
: &lt;math&gt;(A\smallsetminus B)\cap (B\smallsetminus A)&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', consider the equations
: &lt;math&gt; A\cup B = (A\smallsetminus B)\cup B&lt;/math&gt;
: &lt;math&gt;A\cup B = (A\smallsetminus B)\cup (B\smallsetminus A)&lt;/math&gt;

For each equation, is it always true?

Is it sometimes true?
{{robelbox/close}}
=== Universal Set and Complement ===
In many settings, we will be exclusively interested in sets which are all subsets of a single set.  

For instance, in some settings we will only be interested in the natural numbers, and all of the sets that we consider will be subsets of it.

Or we might only be interested in the set of real numbers, or other kinds of sets, like the set of all matrices.

Whenever we restrict our interests to the subsets of one particular set, we call that one set the &quot;universal set&quot;.  We may choose the universal set to be any set that we like.

Once we specify a universal set, we can then define the notion of &quot;the complement of a set&quot;.

Suppose that we choose the universal set &lt;math&gt;U=\{0,1,2,3,4\}&lt;/math&gt; and consider the subset &lt;math&gt;A=\{2,3\}&lt;/math&gt;.  Then the complement of ''A'' is the set of all elements in the universe which are ''not'' in ''A''.
: &lt;math&gt;A^c = \{0,1,4\}&lt;/math&gt;

Notice that the choice of universal set will affect the complement of a given set.  If the universal set is &lt;math&gt;U=\{1,2,3,4\}&lt;/math&gt; and &lt;math&gt;A=\{2,3\}&lt;/math&gt; then 
: &lt;math&gt;A^c = \{1,4\}&lt;/math&gt;
{{robelbox|title=Exercise|theme=2}}
Show that for any universal set ''U'' and subset &lt;math&gt;A\subseteq U&lt;/math&gt;, 
: &lt;math&gt;A^c = U\smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the universal set &lt;math&gt;U=\Bbb N&lt;/math&gt; and subsets &lt;math&gt;A=\{2,4,6,\dots\}&lt;/math&gt; and &lt;math&gt;B = \{1,2\}&lt;/math&gt;.

Find the following sets.

(1.) &lt;math&gt;A^c&lt;/math&gt;

(2.) &lt;math&gt;B^c&lt;/math&gt;

(3.) &lt;math&gt;A^c \cap B^c&lt;/math&gt;

(4.) &lt;math&gt;(A\cup B)^c&lt;/math&gt;
{{robelbox/close}}
We can now state an important relationship between the union, intersection, and complement.  
{{robelbox|title=Theorem: U-C De Morgan's}}
''Theorem'': Suppose the universal set is ''U'' with subsets ''A'' and ''B''.  Then 
: &lt;math&gt;(A\cup B)^c = A^c \cap B^c&lt;/math&gt;
----
In order to prove that two sets are equal, we must prove that every element in the left set is in the right set; and every element in the right set is in the left set.

In short, the fundamental way to prove a set equality, &lt;math&gt;X=Y&lt;/math&gt;, is to prove two subset relations.

: &lt;math&gt;X\subseteq Y&lt;/math&gt;
and 
: &lt;math&gt;Y\subseteq X&lt;/math&gt;
We now do just that, with ''X'' identified with the left side, &lt;math&gt;(A\cup B)^c&lt;/math&gt;.  ''Y'' is identified with the right side, &lt;math&gt;A^c\cap B^c&lt;/math&gt;.

''Proof'':

(For &lt;math&gt;(A\cup B)^c\subseteq A^c \cap B^c&lt;/math&gt;.)

{{small|(1.)}} Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.  

(The goal then is to show &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.)
----
{{small|(2.)}} So &lt;math&gt; x\notin A\cup B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (2.) and the definition of union.
----
{{small|(4.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (3.) by logical inference.
----
{{small|(5.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (3.) and the definition of complement.
----
{{small|(6.)}} So &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

''Reason'': From (4.) and the definition of intersection.  
----
{{small|(7.)}} So &lt;math&gt;(A\cup B)^c\subseteq A^c\cap B^c&lt;/math&gt;.

''Reason'': Lines (1.) to (5.) and the definition of subsets.
----
{{small|(8.)}} (For &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.)

Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

(The goal then is to show &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.
----
{{small|(9.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (7.) and the definition of intersection.
----
{{small|(10.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (8.) and the definition of the complement.
----
{{small|(11.)}} So &lt;math&gt;x\notin A\cup B&lt;/math&gt;.

''Reason'': From (9.) and the definition of union.
----
{{small|(12.)}} So &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.

''Reason'': From (10.) and the definition of complement.
----
{{small|(13.)}} So &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.

''Reason'': From (7.) to (11.) and the definition of subsets.
----
{{small|(14.)}} So &lt;math&gt;(A\cup B)^c = A^c\cap B^c&lt;/math&gt;.

''Reason'': From (6.) and (12.) and the definition of set equality.
{{robelbox/close}}

Notice that the way that this proof flows, is to start from formal statements, like &quot;Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot; or &quot;Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.&quot;

From there, we progressively &quot;unpack&quot; this formalism, into a language which is more like a logical or natural language.  For example, from &quot;&lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot;, we eventually arrive at &quot;&lt;math&gt;x\notin A\cup B&lt;/math&gt;&quot; and then &quot;''x'' is not in ''A'' or ''B''&quot;.

At this point we have almost entirely exchanged the formalism for logical expressions.  Complements (formal) have all been exchanged for negations (logical).  Union (formal) has been exchange for disjunction (logical).  

Here we are free to reason logically, and we use that freedom to reason that &quot;''x'' is not in ''A'' or ''B''&quot; is logically equivalent to &quot;''x'' is not in ''A'' and ''x'' is not in ''B''&quot;.  

After this, we return everything to formalism.  At the end of this sequence, we ultimately arrive at the goal: &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.</text>
      <sha1>eklso4svezg3jopx7f263d8peb6ufz6</sha1>
    </revision>
    <revision>
      <id>2622950</id>
      <parentid>2622948</parentid>
      <timestamp>2024-04-26T03:22:28Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Universal Set and Complement */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10485" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Can we solve set equations the way that we solve algebraic equations?

Suppose that ''X'' is some set which satisfies the equation 
: &lt;math&gt;X \cup \{1,2,3\}=\{-2,1,2,3,4\}&lt;/math&gt;

Can we determine what ''X'' must be?

If we cannot infer what ''X'' is, then can we at least make some inference about the elements of ''X''?
{{robelbox/close}}

=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', find 
: &lt;math&gt;(A\smallsetminus B)\cap (B\smallsetminus A)&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', consider the equations
: &lt;math&gt; A\cup B = (A\smallsetminus B)\cup B&lt;/math&gt;
: &lt;math&gt;A\cup B = (A\smallsetminus B)\cup (B\smallsetminus A)&lt;/math&gt;

For each equation, is it always true?

Is it sometimes true?
{{robelbox/close}}
=== Universal Set and Complement ===
In many settings, we will be exclusively interested in sets which are all subsets of a single set.  

For instance, in some settings we will only be interested in the natural numbers, and all of the sets that we consider will be subsets of it.

Or we might only be interested in the set of real numbers, or other kinds of sets, like the set of all matrices.

Whenever we restrict our interests to the subsets of one particular set, we call that one set the &quot;universal set&quot;.  We may choose the universal set to be any set that we like.

Once we specify a universal set, we can then define the notion of &quot;the complement of a set&quot;.

Suppose that we choose the universal set &lt;math&gt;U=\{0,1,2,3,4\}&lt;/math&gt; and consider the subset &lt;math&gt;A=\{2,3\}&lt;/math&gt;.  Then the complement of ''A'' is the set of all elements in the universe which are ''not'' in ''A''.
: &lt;math&gt;A^c = \{0,1,4\}&lt;/math&gt;

Notice that the choice of universal set will affect the complement of a given set.  If the universal set is &lt;math&gt;U=\{1,2,3,4\}&lt;/math&gt; and &lt;math&gt;A=\{2,3\}&lt;/math&gt; then 
: &lt;math&gt;A^c = \{1,4\}&lt;/math&gt;
{{robelbox|title=Exercise|theme=2}}
Show that for any universal set ''U'' and subset &lt;math&gt;A\subseteq U&lt;/math&gt;, 
: &lt;math&gt;A^c = U\smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the universal set &lt;math&gt;U=\Bbb N&lt;/math&gt; and subsets &lt;math&gt;A=\{2,4,6,\dots\}&lt;/math&gt; and &lt;math&gt;B = \{1,2\}&lt;/math&gt;.

Find the following sets.

(1.) &lt;math&gt;A^c&lt;/math&gt;

(2.) &lt;math&gt;B^c&lt;/math&gt;

(3.) &lt;math&gt;A^c \cap B^c&lt;/math&gt;

(4.) &lt;math&gt;(A\cup B)^c&lt;/math&gt;
{{robelbox/close}}
We can now state an important relationship between the union, intersection, and complement.  
{{robelbox|title=Theorem: U-C De Morgan's}}
''Theorem'': Suppose the universal set is ''U'' with subsets ''A'' and ''B''.  Then 
: &lt;math&gt;(A\cup B)^c = A^c \cap B^c&lt;/math&gt;
----
In order to prove that two sets are equal, we must prove that every element in the left set is in the right set; and every element in the right set is in the left set.

In short, the fundamental way to prove a set equality, &lt;math&gt;X=Y&lt;/math&gt;, is to prove two subset relations.

: &lt;math&gt;X\subseteq Y&lt;/math&gt;
and 
: &lt;math&gt;Y\subseteq X&lt;/math&gt;
We now do just that, with ''X'' identified with the left side, &lt;math&gt;(A\cup B)^c&lt;/math&gt;.  ''Y'' is identified with the right side, &lt;math&gt;A^c\cap B^c&lt;/math&gt;.

''Proof'':

(For &lt;math&gt;(A\cup B)^c\subseteq A^c \cap B^c&lt;/math&gt;.)

{{small|(1.)}} Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.  

(The goal then is to show &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.)
----
{{small|(2.)}} So &lt;math&gt; x\notin A\cup B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (2.) and the definition of union.
----
{{small|(4.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (3.) by logical inference.
----
{{small|(5.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (4.) and the definition of complement.
----
{{small|(6.)}} So &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

''Reason'': From (5.) and the definition of intersection.  
----
{{small|(7.)}} So &lt;math&gt;(A\cup B)^c\subseteq A^c\cap B^c&lt;/math&gt;.

''Reason'': Lines (1.) to (6.) and the definition of subsets.
----
{{small|(8.)}} (For &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.)

Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

(The goal then is to show &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.
----
{{small|(9.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (8.) and the definition of intersection.
----
{{small|(10.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (9.) and the definition of the complement.
----
{{small|(11.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (10.) and logical inference.
{{small|(12.)}} So &lt;math&gt;x\notin A\cup B&lt;/math&gt;.

''Reason'': From (11.) and the definition of union.
----
{{small|(13.)}} So &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.

''Reason'': From (12.) and the definition of complement.
----
{{small|(14.)}} So &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.

''Reason'': From (8.) to (13.) and the definition of subsets.
----
{{small|(15.)}} So &lt;math&gt;(A\cup B)^c = A^c\cap B^c&lt;/math&gt;.

''Reason'': From (7.) and (14.) and the definition of set equality.
{{robelbox/close}}

Notice that the way that this proof flows, is to start from formal statements, like &quot;Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot; or &quot;Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.&quot;

From there, we progressively &quot;unpack&quot; this formalism, into a language which is more like a logical or natural language.  For example, from &quot;&lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot;, we eventually arrive at &quot;&lt;math&gt;x\notin A\cup B&lt;/math&gt;&quot; and then &quot;''x'' is not in ''A'' or ''B''&quot;.

At this point we have almost entirely exchanged the formalism for logical expressions.  Complements (formal) have all been exchanged for negations (logical).  Union (formal) has been exchange for disjunction (logical).  

Now that we have exchanged formalism for logical expression, we are free to reason logically.  We use that freedom to reason that &quot;''x'' is not in ''A'' or ''B''&quot; is logically equivalent to &quot;''x'' is not in ''A'' and ''x'' is not in ''B''&quot;.  

After this, we return everything to formalism.  At the end of this sequence, we ultimately arrive at the goal: &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

In the following proof, we repeat a very similar flow.  However, the logic required is a bit different from the above proof.
{{robelbox|title=Theorem: I-C De Morgan's}}
''Theorem'': Let ''A'' and ''B'' be two sets with universe ''U''.  Then 
: &lt;math&gt;(A\cap B)^c= A^c\cup B^c&lt;/math&gt;
----
''Proof''

{{small|(1.)}}
{{robelbox/close}}</text>
      <sha1>rqjx6rr9aciqo2xxt4b30hly11sa1kb</sha1>
    </revision>
    <revision>
      <id>2622951</id>
      <parentid>2622950</parentid>
      <timestamp>2024-04-26T03:37:28Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Universal Set and Complement */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10940" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Can we solve set equations the way that we solve algebraic equations?

Suppose that ''X'' is some set which satisfies the equation 
: &lt;math&gt;X \cup \{1,2,3\}=\{-2,1,2,3,4\}&lt;/math&gt;

Can we determine what ''X'' must be?

If we cannot infer what ''X'' is, then can we at least make some inference about the elements of ''X''?
{{robelbox/close}}

=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', find 
: &lt;math&gt;(A\smallsetminus B)\cap (B\smallsetminus A)&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', consider the equations
: &lt;math&gt; A\cup B = (A\smallsetminus B)\cup B&lt;/math&gt;
: &lt;math&gt;A\cup B = (A\smallsetminus B)\cup (B\smallsetminus A)&lt;/math&gt;

For each equation, is it always true?

Is it sometimes true?
{{robelbox/close}}
=== Universal Set and Complement ===
In many settings, we will be exclusively interested in sets which are all subsets of a single set.  

For instance, in some settings we will only be interested in the natural numbers, and all of the sets that we consider will be subsets of it.

Or we might only be interested in the set of real numbers, or other kinds of sets, like the set of all matrices.

Whenever we restrict our interests to the subsets of one particular set, we call that one set the &quot;universal set&quot;.  We may choose the universal set to be any set that we like.

Once we specify a universal set, we can then define the notion of &quot;the complement of a set&quot;.

Suppose that we choose the universal set &lt;math&gt;U=\{0,1,2,3,4\}&lt;/math&gt; and consider the subset &lt;math&gt;A=\{2,3\}&lt;/math&gt;.  Then the complement of ''A'' is the set of all elements in the universe which are ''not'' in ''A''.
: &lt;math&gt;A^c = \{0,1,4\}&lt;/math&gt;

Notice that the choice of universal set will affect the complement of a given set.  If the universal set is &lt;math&gt;U=\{1,2,3,4\}&lt;/math&gt; and &lt;math&gt;A=\{2,3\}&lt;/math&gt; then 
: &lt;math&gt;A^c = \{1,4\}&lt;/math&gt;
{{robelbox|title=Exercise|theme=2}}
Show that for any universal set ''U'' and subset &lt;math&gt;A\subseteq U&lt;/math&gt;, 
: &lt;math&gt;A^c = U\smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the universal set &lt;math&gt;U=\Bbb N&lt;/math&gt; and subsets &lt;math&gt;A=\{2,4,6,\dots\}&lt;/math&gt; and &lt;math&gt;B = \{1,2\}&lt;/math&gt;.

Find the following sets.

(1.) &lt;math&gt;A^c&lt;/math&gt;

(2.) &lt;math&gt;B^c&lt;/math&gt;

(3.) &lt;math&gt;A^c \cap B^c&lt;/math&gt;

(4.) &lt;math&gt;(A\cup B)^c&lt;/math&gt;
{{robelbox/close}}
We can now state an important relationship between the union, intersection, and complement.  
{{robelbox|title=Theorem: U-C De Morgan's}}
''Theorem'': Suppose the universal set is ''U'' with subsets ''A'' and ''B''.  Then 
: &lt;math&gt;(A\cup B)^c = A^c \cap B^c&lt;/math&gt;
----
In order to prove that two sets are equal, we must prove that every element in the left set is in the right set; and every element in the right set is in the left set.

In short, the fundamental way to prove a set equality, &lt;math&gt;X=Y&lt;/math&gt;, is to prove two subset relations.

: &lt;math&gt;X\subseteq Y&lt;/math&gt;
and 
: &lt;math&gt;Y\subseteq X&lt;/math&gt;
We now do just that, with ''X'' identified with the left side, &lt;math&gt;(A\cup B)^c&lt;/math&gt;.  ''Y'' is identified with the right side, &lt;math&gt;A^c\cap B^c&lt;/math&gt;.

''Proof'':

(For &lt;math&gt;(A\cup B)^c\subseteq A^c \cap B^c&lt;/math&gt;.)

{{small|(1.)}} Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.  

(The goal then is to show &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.)
----
{{small|(2.)}} So &lt;math&gt; x\notin A\cup B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (2.) and the definition of union.
----
{{small|(4.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (3.) by logical inference.
----
{{small|(5.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (4.) and the definition of complement.
----
{{small|(6.)}} So &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

''Reason'': From (5.) and the definition of intersection.  
----
{{small|(7.)}} So &lt;math&gt;(A\cup B)^c\subseteq A^c\cap B^c&lt;/math&gt;.

''Reason'': Lines (1.) to (6.) and the definition of subsets.
----
{{small|(8.)}} (For &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.)

Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

(The goal then is to show &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.
----
{{small|(9.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (8.) and the definition of intersection.
----
{{small|(10.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (9.) and the definition of the complement.
----
{{small|(11.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (10.) and logical inference.
{{small|(12.)}} So &lt;math&gt;x\notin A\cup B&lt;/math&gt;.

''Reason'': From (11.) and the definition of union.
----
{{small|(13.)}} So &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.

''Reason'': From (12.) and the definition of complement.
----
{{small|(14.)}} So &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.

''Reason'': From (8.) to (13.) and the definition of subsets.
----
{{small|(15.)}} So &lt;math&gt;(A\cup B)^c = A^c\cap B^c&lt;/math&gt;.

''Reason'': From (7.) and (14.) and the definition of set equality.
{{robelbox/close}}

Notice that the way that this proof flows, is to start from formal statements, like &quot;Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot; or &quot;Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.&quot;

From there, we progressively &quot;unpack&quot; this formalism, into a language which is more like a logical or natural language.  For example, from &quot;&lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot;, we eventually arrive at &quot;&lt;math&gt;x\notin A\cup B&lt;/math&gt;&quot; and then &quot;''x'' is not in ''A'' or ''B''&quot;.

At this point we have almost entirely exchanged the formalism for logical expressions.  Complements (formal) have all been exchanged for negations (logical).  Union (formal) has been exchange for disjunction (logical).  

Now that we have exchanged formalism for logical expression, we are free to reason logically.  We use that freedom to reason that &quot;''x'' is not in ''A'' or ''B''&quot; is logically equivalent to &quot;''x'' is not in ''A'' and ''x'' is not in ''B''&quot;.  

After this, we return everything to formalism.  At the end of this sequence, we ultimately arrive at the goal: &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

In the following proof, we repeat a very similar flow.  However, the logic required is a bit different from the above proof.
{{robelbox|title=Theorem: I-C De Morgan's}}
''Theorem'': Let ''A'' and ''B'' be two sets with universe ''U''.  Then 
: &lt;math&gt;(A\cap B)^c= A^c\cup B^c&lt;/math&gt;
----
As is common with set equalities, we again prove to subset relations.

''Proof'':

{{small|(1.)}}  Let &lt;math&gt;x\in (A\cap B)^c&lt;/math&gt;.
----
{{small|(2.)}} So &lt;math&gt;x\notin A\cap B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' and ''B''.

''Reason'': From (2.) and the definition of intersection.
----
{{small|(4.)}} So ''x'' is either not ''A'', or ''x'' is not in ''B''.

''Reason'': From (3.) and logical inference.
{{robelbox/close}}</text>
      <sha1>gy1c3wbxa722nsrqhkd6hl6j78lhue5</sha1>
    </revision>
    <revision>
      <id>2622953</id>
      <parentid>2622951</parentid>
      <timestamp>2024-04-26T03:53:12Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Universal Set and Complement */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12290" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Can we solve set equations the way that we solve algebraic equations?

Suppose that ''X'' is some set which satisfies the equation 
: &lt;math&gt;X \cup \{1,2,3\}=\{-2,1,2,3,4\}&lt;/math&gt;

Can we determine what ''X'' must be?

If we cannot infer what ''X'' is, then can we at least make some inference about the elements of ''X''?
{{robelbox/close}}

=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', find 
: &lt;math&gt;(A\smallsetminus B)\cap (B\smallsetminus A)&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', consider the equations
: &lt;math&gt; A\cup B = (A\smallsetminus B)\cup B&lt;/math&gt;
: &lt;math&gt;A\cup B = (A\smallsetminus B)\cup (B\smallsetminus A)&lt;/math&gt;

For each equation, is it always true?

Is it sometimes true?
{{robelbox/close}}
=== Universal Set and Complement ===
In many settings, we will be exclusively interested in sets which are all subsets of a single set.  

For instance, in some settings we will only be interested in the natural numbers, and all of the sets that we consider will be subsets of it.

Or we might only be interested in the set of real numbers, or other kinds of sets, like the set of all matrices.

Whenever we restrict our interests to the subsets of one particular set, we call that one set the &quot;universal set&quot;.  We may choose the universal set to be any set that we like.

Once we specify a universal set, we can then define the notion of &quot;the complement of a set&quot;.

Suppose that we choose the universal set &lt;math&gt;U=\{0,1,2,3,4\}&lt;/math&gt; and consider the subset &lt;math&gt;A=\{2,3\}&lt;/math&gt;.  Then the complement of ''A'' is the set of all elements in the universe which are ''not'' in ''A''.
: &lt;math&gt;A^c = \{0,1,4\}&lt;/math&gt;

Notice that the choice of universal set will affect the complement of a given set.  If the universal set is &lt;math&gt;U=\{1,2,3,4\}&lt;/math&gt; and &lt;math&gt;A=\{2,3\}&lt;/math&gt; then 
: &lt;math&gt;A^c = \{1,4\}&lt;/math&gt;
{{robelbox|title=Exercise|theme=2}}
Show that for any universal set ''U'' and subset &lt;math&gt;A\subseteq U&lt;/math&gt;, 
: &lt;math&gt;A^c = U\smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the universal set &lt;math&gt;U=\Bbb N&lt;/math&gt; and subsets &lt;math&gt;A=\{2,4,6,\dots\}&lt;/math&gt; and &lt;math&gt;B = \{1,2\}&lt;/math&gt;.

Find the following sets.

(1.) &lt;math&gt;A^c&lt;/math&gt;

(2.) &lt;math&gt;B^c&lt;/math&gt;

(3.) &lt;math&gt;A^c \cap B^c&lt;/math&gt;

(4.) &lt;math&gt;(A\cup B)^c&lt;/math&gt;
{{robelbox/close}}
We can now state an important relationship between the union, intersection, and complement.  
{{robelbox|title=Theorem: U-C De Morgan's}}
''Theorem'': Suppose the universal set is ''U'' with subsets ''A'' and ''B''.  Then 
: &lt;math&gt;(A\cup B)^c = A^c \cap B^c&lt;/math&gt;
----
In order to prove that two sets are equal, we must prove that every element in the left set is in the right set; and every element in the right set is in the left set.

In short, the fundamental way to prove a set equality, &lt;math&gt;X=Y&lt;/math&gt;, is to prove two subset relations.

: &lt;math&gt;X\subseteq Y&lt;/math&gt;
and 
: &lt;math&gt;Y\subseteq X&lt;/math&gt;
We now do just that, with ''X'' identified with the left side, &lt;math&gt;(A\cup B)^c&lt;/math&gt;.  ''Y'' is identified with the right side, &lt;math&gt;A^c\cap B^c&lt;/math&gt;.

''Proof'':

(For &lt;math&gt;(A\cup B)^c\subseteq A^c \cap B^c&lt;/math&gt;.)

{{small|(1.)}} Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.  

(The goal then is to show &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.)
----
{{small|(2.)}} So &lt;math&gt; x\notin A\cup B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (2.) and the definition of union.
----
{{small|(4.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (3.) by logical inference.
----
{{small|(5.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (4.) and the definition of complement.
----
{{small|(6.)}} So &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

''Reason'': From (5.) and the definition of intersection.  
----
{{small|(7.)}} So &lt;math&gt;(A\cup B)^c\subseteq A^c\cap B^c&lt;/math&gt;.

''Reason'': Lines (1.) to (6.) and the definition of subsets.
----
{{small|(8.)}} (For &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.)

Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

(The goal then is to show &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.
----
{{small|(9.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (8.) and the definition of intersection.
----
{{small|(10.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (9.) and the definition of the complement.
----
{{small|(11.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (10.) and logical inference.
{{small|(12.)}} So &lt;math&gt;x\notin A\cup B&lt;/math&gt;.

''Reason'': From (11.) and the definition of union.
----
{{small|(13.)}} So &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.

''Reason'': From (12.) and the definition of complement.
----
{{small|(14.)}} So &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.

''Reason'': From (8.) to (13.) and the definition of subsets.
----
{{small|(15.)}} So &lt;math&gt;(A\cup B)^c = A^c\cap B^c&lt;/math&gt;.

''Reason'': From (7.) and (14.) and the definition of set equality.
{{robelbox/close}}

Notice that the way that this proof flows, is to start from formal statements, like &quot;Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot; or &quot;Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.&quot;

From there, we progressively &quot;unpack&quot; this formalism, into a language which is more like a logical or natural language.  For example, from &quot;&lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot;, we eventually arrive at &quot;&lt;math&gt;x\notin A\cup B&lt;/math&gt;&quot; and then &quot;''x'' is not in ''A'' or ''B''&quot;.

At this point we have almost entirely exchanged the formalism for logical expressions.  Complements (formal) have all been exchanged for negations (logical).  Union (formal) has been exchange for disjunction (logical).  

Now that we have exchanged formalism for logical expression, we are free to reason logically.  We use that freedom to reason that &quot;''x'' is not in ''A'' or ''B''&quot; is logically equivalent to &quot;''x'' is not in ''A'' and ''x'' is not in ''B''&quot;.  

After this, we return everything to formalism.  At the end of this sequence, we ultimately arrive at the goal: &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

In the following proof, we repeat a very similar flow.  However, the logic required is a bit different from the above proof.
{{robelbox|title=Theorem: I-C De Morgan's}}
''Theorem'': Let ''A'' and ''B'' be two sets with universe ''U''.  Then 
: &lt;math&gt;(A\cap B)^c= A^c\cup B^c&lt;/math&gt;
----
As is common with set equalities, we again prove to subset relations.

''Proof'':

{{small|(1.)}}  Let &lt;math&gt;x\in (A\cap B)^c&lt;/math&gt;.
----
{{small|(2.)}} So &lt;math&gt;x\notin A\cap B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' and ''B''.

''Reason'': From (2.) and the definition of intersection.
----
{{small|(4.)}} So either ''x'' is not ''A'', or ''x'' is not in ''B''.

''Reason'': From (3.) and logical inference.
----
{{small|(5.)}} Assume ''x'' is not in ''A''.

(This is the start of a proof by cases.  In outline, we will assume ''x'' is not in ''A'' and then prove &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.  Then we will assume ''x'' is not in ''B'' and then again prove &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.  Because of (4.) we will therefore conclude that, in every case, &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.)
----
{{small|(6.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt;.

''Reason'': From (5.) and the definition of complement.
----
{{small|(7.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt; or &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (6.) and logical inference.
----
{{small|(8.)}} Then &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': From (7.) by definition of union.
----
{{small|(9.)}} Now assume &lt;math&gt;x\in B^c&lt;/math&gt;.

(We have finished the first case with line (8.) and this begins the second case.)
----
{{small|(10.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt; or &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (9.) by logical inference.
----
{{small|(11.)}} Then &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': From (10.) by definition of union.
----
{{small|(12.)}} Therefore &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': Proof by cases, from (4.) and (8.) and (11.).
---- 
{{small|(13.)}} So &lt;math&gt;(A\cap B)^c \subseteq A^c\cup B^c&lt;/math&gt;.

''Reason'': From (1.) through (12.) and the definition of subsets.
----
{{small|(14.)}}
{{robelbox/close}}</text>
      <sha1>9a7pkxh6brjjwp0gy27jbqpxet8mw10</sha1>
    </revision>
    <revision>
      <id>2623019</id>
      <parentid>2622953</parentid>
      <timestamp>2024-04-26T11:30:53Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Universal Set and Complement */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13921" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Can we solve set equations the way that we solve algebraic equations?

Suppose that ''X'' is some set which satisfies the equation 
: &lt;math&gt;X \cup \{1,2,3\}=\{-2,1,2,3,4\}&lt;/math&gt;

Can we determine what ''X'' must be?

If we cannot infer what ''X'' is, then can we at least make some inference about the elements of ''X''?
{{robelbox/close}}

=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', find 
: &lt;math&gt;(A\smallsetminus B)\cap (B\smallsetminus A)&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', consider the equations
: &lt;math&gt; A\cup B = (A\smallsetminus B)\cup B&lt;/math&gt;
: &lt;math&gt;A\cup B = (A\smallsetminus B)\cup (B\smallsetminus A)&lt;/math&gt;

For each equation, is it always true?

Is it sometimes true?
{{robelbox/close}}
=== Universal Set and Complement ===
[[File:PolygonsSetComplement.svg|thumb|A pictorial representation of the complement of a set.]]
In many settings, we will be exclusively interested in sets which are all subsets of a single set.  

For instance, in some settings we will only be interested in the natural numbers, and all of the sets that we consider will be subsets of it.

Or we might only be interested in the set of real numbers, or other kinds of sets, like the set of all matrices.

Whenever we restrict our interests to the subsets of one particular set, we call that one set the &quot;universal set&quot;.  We may choose the universal set to be any set that we like.

Once we specify a universal set, we can then define the notion of &quot;the complement of a set&quot;.

Suppose that we choose the universal set &lt;math&gt;U=\{0,1,2,3,4\}&lt;/math&gt; and consider the subset &lt;math&gt;A=\{2,3\}&lt;/math&gt;.  Then the complement of ''A'' is the set of all elements in the universe which are ''not'' in ''A''.
: &lt;math&gt;A^c = \{0,1,4\}&lt;/math&gt;

Notice that the choice of universal set will affect the complement of a given set.  If the universal set is &lt;math&gt;U=\{1,2,3,4\}&lt;/math&gt; and &lt;math&gt;A=\{2,3\}&lt;/math&gt; then 
: &lt;math&gt;A^c = \{1,4\}&lt;/math&gt;
{{robelbox|title=Exercise|theme=2}}
Show that for any universal set ''U'' and subset &lt;math&gt;A\subseteq U&lt;/math&gt;, 
: &lt;math&gt;A^c = U\smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the universal set &lt;math&gt;U=\Bbb N&lt;/math&gt; and subsets &lt;math&gt;A=\{2,4,6,\dots\}&lt;/math&gt; and &lt;math&gt;B = \{1,2\}&lt;/math&gt;.

Find the following sets.

(1.) &lt;math&gt;A^c&lt;/math&gt;

(2.) &lt;math&gt;B^c&lt;/math&gt;

(3.) &lt;math&gt;A^c \cap B^c&lt;/math&gt;

(4.) &lt;math&gt;(A\cup B)^c&lt;/math&gt;
{{robelbox/close}}
We can now state an important relationship between the union, intersection, and complement.  
{{robelbox|title=Theorem: U-C De Morgan's}}
''Theorem'': Suppose the universal set is ''U'' with subsets ''A'' and ''B''.  Then 
: &lt;math&gt;(A\cup B)^c = A^c \cap B^c&lt;/math&gt;
----
In order to prove that two sets are equal, we must prove that every element in the left set is in the right set; and every element in the right set is in the left set.

In short, the fundamental way to prove a set equality, &lt;math&gt;X=Y&lt;/math&gt;, is to prove two subset relations.

: &lt;math&gt;X\subseteq Y&lt;/math&gt;
and 
: &lt;math&gt;Y\subseteq X&lt;/math&gt;
We now do just that, with ''X'' identified with the left side, &lt;math&gt;(A\cup B)^c&lt;/math&gt;.  ''Y'' is identified with the right side, &lt;math&gt;A^c\cap B^c&lt;/math&gt;.

''Proof'':

(For &lt;math&gt;(A\cup B)^c\subseteq A^c \cap B^c&lt;/math&gt;.)

{{small|(1.)}} Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.  

(The goal then is to show &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.)
----
{{small|(2.)}} So &lt;math&gt; x\notin A\cup B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (2.) and the definition of union.
----
{{small|(4.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (3.) by logical inference.
----
{{small|(5.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (4.) and the definition of complement.
----
{{small|(6.)}} So &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

''Reason'': From (5.) and the definition of intersection.  
----
{{small|(7.)}} So &lt;math&gt;(A\cup B)^c\subseteq A^c\cap B^c&lt;/math&gt;.

''Reason'': Lines (1.) to (6.) and the definition of subsets.
----
{{small|(8.)}} (For &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.)

Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

(The goal then is to show &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.
----
{{small|(9.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (8.) and the definition of intersection.
----
{{small|(10.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (9.) and the definition of the complement.
----
{{small|(11.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (10.) and logical inference.
{{small|(12.)}} So &lt;math&gt;x\notin A\cup B&lt;/math&gt;.

''Reason'': From (11.) and the definition of union.
----
{{small|(13.)}} So &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.

''Reason'': From (12.) and the definition of complement.
----
{{small|(14.)}} So &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.

''Reason'': From (8.) to (13.) and the definition of subsets.
----
{{small|(15.)}} So &lt;math&gt;(A\cup B)^c = A^c\cap B^c&lt;/math&gt;.

''Reason'': From (7.) and (14.) and the definition of set equality.
{{robelbox/close}}

[[File:Algebra1 ins fig017 dem.svg|thumb|A representation of the I-C De Morgan's law.  The notation &lt;math&gt;\overline A&lt;/math&gt; is an alternate notation for the complement, &lt;math&gt;A^c&lt;/math&gt;.]]

Notice that the way that this proof flows, is to start from formal statements, like &quot;Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot; or &quot;Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.&quot;

From there, we progressively &quot;unpack&quot; this formalism, into a language which is more like a logical or natural language.  For example, from &quot;&lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot;, we eventually arrive at &quot;&lt;math&gt;x\notin A\cup B&lt;/math&gt;&quot; and then &quot;''x'' is not in ''A'' or ''B''&quot;.

At this point we have almost entirely exchanged the formalism for logical expressions.  Complements (formal) have all been exchanged for negations (logical).  Union (formal) has been exchange for disjunction (logical).  

Now that we have exchanged formalism for logical expression, we are free to reason logically.  We use that freedom to reason that &quot;''x'' is not in ''A'' or ''B''&quot; is logically equivalent to &quot;''x'' is not in ''A'' and ''x'' is not in ''B''&quot;.  

After this, we return everything to formalism.  At the end of this sequence, we ultimately arrive at the goal: &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

In the following proof, we repeat a very similar flow.  However, the logic required is a bit different from the above proof.
{{robelbox|title=Theorem: I-C De Morgan's}}
''Theorem'': Let ''A'' and ''B'' be two sets with universe ''U''.  Then 
: &lt;math&gt;(A\cap B)^c= A^c\cup B^c&lt;/math&gt;
----
As is common with set equalities, we again prove to subset relations.

''Proof'':

{{small|(1.)}}  Let &lt;math&gt;x\in (A\cap B)^c&lt;/math&gt;.
----
{{small|(2.)}} So &lt;math&gt;x\notin A\cap B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' and ''B''.

''Reason'': From (2.) and the definition of intersection.
----
{{small|(4.)}} So either ''x'' is not ''A'', or ''x'' is not in ''B''.

''Reason'': From (3.) and logical inference.
----
{{small|(5.)}} Assume ''x'' is not in ''A''.

(This is the start of a proof by cases.  In outline, we will assume ''x'' is not in ''A'' and then prove &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.  Then we will assume ''x'' is not in ''B'' and then again prove &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.  Because of (4.) we will therefore conclude that, in every case, &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.)
----
{{small|(6.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt;.

''Reason'': From (5.) and the definition of complement.
----
{{small|(7.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt; or &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (6.) and logical inference.
----
{{small|(8.)}} Then &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': From (7.) by definition of union.
----
{{small|(9.)}} Now assume &lt;math&gt;x\in B^c&lt;/math&gt;.

(We have finished the first case with line (8.) and this begins the second case.)
----
{{small|(10.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt; or &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (9.) by logical inference.
----
{{small|(11.)}} Then &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': From (10.) by definition of union.
----
{{small|(12.)}} Therefore &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': Proof by cases, from (4.) and (8.) and (11.).
---- 
{{small|(13.)}} So &lt;math&gt;(A\cap B)^c \subseteq A^c\cup B^c&lt;/math&gt;.

''Reason'': From (1.) through (12.) and the definition of subsets.
----
{{small|(14.)}} Let &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.
----
{{small|(15.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; or &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (14.) by definition of union.  
----
{{small|(16.)}} Assume (for proof by cases) &lt;math&gt;x\in A^c&lt;/math&gt;.
----
{{small|(17.)}} Then &lt;math&gt;x\notin A&lt;/math&gt;.

''Reason'': From (16.) by the definition of complement.
----
{{small|(18.)}} Then ''x'' not in ''A'' and ''B''.

''Reason'': From (17.) by logical inference.
----
{{small|(19.)}} Then &lt;math&gt;x\notin A\cap B&lt;/math&gt;.

''Reason'': From (18.) by definition of intersection.
----
{{small|(20.)}} Then &lt;math&gt;x\in(A\cap B)^c&lt;/math&gt;.

''Reason'': From (19.) by definition of complement.
----
{{small|(21.)}} Now assume (for the second case, in the proof by cases) that &lt;math&gt;x\in B^c&lt;/math&gt;.
----
{{small|(22.)}} Then &lt;math&gt;x\in (A\cap B)^c&lt;/math&gt;.

''Reason'': Repeat a sequence of steps, ''mutatis mutandis'', like the steps from (16.) to (19.).
----
{{small|(23.)}} Therefore, in all cases, &lt;math&gt;x\in (A\cap B)^c&lt;/math&gt;.

''Reason'': Proof by cases, from (15.) and (16. to 20.) and (21. to 22.).
----
{{small|(24.)}} Therefore &lt;math&gt;A^c\cup B^c\subseteq (A\cap B)^c&lt;/math&gt;.

''Reason'': From (14.) to (23.) by definition of subsets.
----
{{small|(25.)}} Therefore &lt;math&gt;(A\cap B)^c = A^c\cup B^c&lt;/math&gt;.

''Reason'': From (13.) and (25.) by definition of set equality.
{{robelbox/close}}</text>
      <sha1>5240d2h50kl2l7lkq6rlf2e7gdf5hzo</sha1>
    </revision>
    <revision>
      <id>2623020</id>
      <parentid>2623019</parentid>
      <timestamp>2024-04-26T11:36:45Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Universal Set and Complement */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13922" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Can we solve set equations the way that we solve algebraic equations?

Suppose that ''X'' is some set which satisfies the equation 
: &lt;math&gt;X \cup \{1,2,3\}=\{-2,1,2,3,4\}&lt;/math&gt;

Can we determine what ''X'' must be?

If we cannot infer what ''X'' is, then can we at least make some inference about the elements of ''X''?
{{robelbox/close}}

=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', find 
: &lt;math&gt;(A\smallsetminus B)\cap (B\smallsetminus A)&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', consider the equations
: &lt;math&gt; A\cup B = (A\smallsetminus B)\cup B&lt;/math&gt;
: &lt;math&gt;A\cup B = (A\smallsetminus B)\cup (B\smallsetminus A)&lt;/math&gt;

For each equation, is it always true?

Is it sometimes true?
{{robelbox/close}}
=== Universal Set and Complement ===
[[File:PolygonsSetComplement.svg|thumb|A pictorial representation of the complement of a set.]]
In many settings, we will be exclusively interested in sets which are all subsets of a single set.  

For instance, in some settings we will only be interested in the natural numbers, and all of the sets that we consider will be subsets of it.

Or we might only be interested in the set of real numbers, or other kinds of sets, like the set of all matrices.

Whenever we restrict our interests to the subsets of one particular set, we call that one set the &quot;universal set&quot;.  We may choose the universal set to be any set that we like.

Once we specify a universal set, we can then define the notion of &quot;the complement of a set&quot;.

Suppose that we choose the universal set &lt;math&gt;U=\{0,1,2,3,4\}&lt;/math&gt; and consider the subset &lt;math&gt;A=\{2,3\}&lt;/math&gt;.  Then the complement of ''A'' is the set of all elements in the universe which are ''not'' in ''A''.
: &lt;math&gt;A^c = \{0,1,4\}&lt;/math&gt;

Notice that the choice of universal set will affect the complement of a given set.  If the universal set is &lt;math&gt;U=\{1,2,3,4\}&lt;/math&gt; and &lt;math&gt;A=\{2,3\}&lt;/math&gt; then 
: &lt;math&gt;A^c = \{1,4\}&lt;/math&gt;
{{robelbox|title=Exercise|theme=2}}
Show that for any universal set ''U'' and subset &lt;math&gt;A\subseteq U&lt;/math&gt;, 
: &lt;math&gt;A^c = U\smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the universal set &lt;math&gt;U=\Bbb N&lt;/math&gt; and subsets &lt;math&gt;A=\{2,4,6,\dots\}&lt;/math&gt; and &lt;math&gt;B = \{1,2\}&lt;/math&gt;.

Find the following sets.

(1.) &lt;math&gt;A^c&lt;/math&gt;

(2.) &lt;math&gt;B^c&lt;/math&gt;

(3.) &lt;math&gt;A^c \cap B^c&lt;/math&gt;

(4.) &lt;math&gt;(A\cup B)^c&lt;/math&gt;
{{robelbox/close}}
We can now state an important relationship between the union, intersection, and complement.  
{{robelbox|title=Theorem: U-C De Morgan's}}
''Theorem'': Suppose the universal set is ''U'' with subsets ''A'' and ''B''.  Then 
: &lt;math&gt;(A\cup B)^c = A^c \cap B^c&lt;/math&gt;
----
In order to prove that two sets are equal, we must prove that every element in the left set is in the right set; and every element in the right set is in the left set.

In short, the fundamental way to prove a set equality, &lt;math&gt;X=Y&lt;/math&gt;, is to prove two subset relations.

: &lt;math&gt;X\subseteq Y&lt;/math&gt;
and 
: &lt;math&gt;Y\subseteq X&lt;/math&gt;
We now do just that, with ''X'' identified with the left side, &lt;math&gt;(A\cup B)^c&lt;/math&gt;.  ''Y'' is identified with the right side, &lt;math&gt;A^c\cap B^c&lt;/math&gt;.

''Proof'':

(For &lt;math&gt;(A\cup B)^c\subseteq A^c \cap B^c&lt;/math&gt;.)

{{small|(1.)}} Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.  

(The goal then is to show &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.)
----
{{small|(2.)}} So &lt;math&gt; x\notin A\cup B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (2.) and the definition of union.
----
{{small|(4.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (3.) by logical inference.
----
{{small|(5.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (4.) and the definition of complement.
----
{{small|(6.)}} So &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

''Reason'': From (5.) and the definition of intersection.  
----
{{small|(7.)}} So &lt;math&gt;(A\cup B)^c\subseteq A^c\cap B^c&lt;/math&gt;.

''Reason'': Lines (1.) to (6.) and the definition of subsets.
----
{{small|(8.)}} (For &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.)

Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

(The goal then is to show &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.
----
{{small|(9.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (8.) and the definition of intersection.
----
{{small|(10.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (9.) and the definition of the complement.
----
{{small|(11.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (10.) and logical inference.
{{small|(12.)}} So &lt;math&gt;x\notin A\cup B&lt;/math&gt;.

''Reason'': From (11.) and the definition of union.
----
{{small|(13.)}} So &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.

''Reason'': From (12.) and the definition of complement.
----
{{small|(14.)}} So &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.

''Reason'': From (8.) to (13.) and the definition of subsets.
----
{{small|(15.)}} So &lt;math&gt;(A\cup B)^c = A^c\cap B^c&lt;/math&gt;.

''Reason'': From (7.) and (14.) and the definition of set equality.
{{robelbox/close}}

[[File:Algebra1 ins fig017 dem.svg|thumb|A representation of the I-C De Morgan's law.  The notation &lt;math&gt;\overline A&lt;/math&gt; is an alternate notation for the complement, &lt;math&gt;A^c&lt;/math&gt;.]]

Notice that the way that this proof flows, is to start from formal statements, like &quot;Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot; or &quot;Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.&quot;

From there, we progressively &quot;unpack&quot; this formalism, into a language which is more like a logical or natural language.  For example, from &quot;&lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot;, we eventually arrive at &quot;&lt;math&gt;x\notin A\cup B&lt;/math&gt;&quot; and then &quot;''x'' is not in ''A'' or ''B''&quot;.

At this point we have almost entirely exchanged the formalism for logical expressions.  Complements (formal) have all been exchanged for negations (logical).  Union (formal) has been exchange for disjunction (logical).  

Now that we have exchanged formalism for logical expression, we are free to reason logically.  We use that freedom to reason that &quot;''x'' is not in ''A'' or ''B''&quot; is logically equivalent to &quot;''x'' is not in ''A'' and ''x'' is not in ''B''&quot;.  

After this, we return everything to formalism.  At the end of this sequence, we ultimately arrive at the goal: &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

In the following proof, we repeat a very similar flow.  However, the logic required is a bit different from the above proof.
{{robelbox|title=Theorem: I-C De Morgan's}}
''Theorem'': Let ''A'' and ''B'' be two sets with universe ''U''.  Then 
: &lt;math&gt;(A\cap B)^c= A^c\cup B^c&lt;/math&gt;
----
As is common with set equalities, we again prove two subset relations.

''Proof'':

{{small|(1.)}}  Let &lt;math&gt;x\in (A\cap B)^c&lt;/math&gt;.
----
{{small|(2.)}} So &lt;math&gt;x\notin A\cap B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' and ''B''.

''Reason'': From (2.) and the definition of intersection.
----
{{small|(4.)}} So either ''x'' is not ''A'', or ''x'' is not in ''B''.

''Reason'': From (3.) and logical inference.
----
{{small|(5.)}} Assume ''x'' is not in ''A''.

(This is the start of a proof by cases.  In outline, we will assume ''x'' is not in ''A'' and then prove &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.  Then we will assume ''x'' is not in ''B'' and then again prove &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.  Because of (4.) we will therefore conclude that, in every case, &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.)
----
{{small|(6.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt;.

''Reason'': From (5.) and the definition of complement.
----
{{small|(7.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt; or &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (6.) and logical inference.
----
{{small|(8.)}} Then &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': From (7.) by definition of union.
----
{{small|(9.)}} Now assume &lt;math&gt;x\in B^c&lt;/math&gt;.

(We have finished the first case with line (8.) and this begins the second case.)
----
{{small|(10.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt; or &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (9.) by logical inference.
----
{{small|(11.)}} Then &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': From (10.) by definition of union.
----
{{small|(12.)}} Therefore &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': Proof by cases, from (4.) and (8.) and (11.).
---- 
{{small|(13.)}} So &lt;math&gt;(A\cap B)^c \subseteq A^c\cup B^c&lt;/math&gt;.

''Reason'': From (1.) through (12.) and the definition of subsets.
----
{{small|(14.)}} Let &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.
----
{{small|(15.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; or &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (14.) by definition of union.  
----
{{small|(16.)}} Assume (for proof by cases) &lt;math&gt;x\in A^c&lt;/math&gt;.
----
{{small|(17.)}} Then &lt;math&gt;x\notin A&lt;/math&gt;.

''Reason'': From (16.) by the definition of complement.
----
{{small|(18.)}} Then ''x'' not in ''A'' and ''B''.

''Reason'': From (17.) by logical inference.
----
{{small|(19.)}} Then &lt;math&gt;x\notin A\cap B&lt;/math&gt;.

''Reason'': From (18.) by definition of intersection.
----
{{small|(20.)}} Then &lt;math&gt;x\in(A\cap B)^c&lt;/math&gt;.

''Reason'': From (19.) by definition of complement.
----
{{small|(21.)}} Now assume (for the second case, in the proof by cases) that &lt;math&gt;x\in B^c&lt;/math&gt;.
----
{{small|(22.)}} Then &lt;math&gt;x\in (A\cap B)^c&lt;/math&gt;.

''Reason'': Repeat a sequence of steps, ''mutatis mutandis'', like the steps from (16.) to (19.).
----
{{small|(23.)}} Therefore, in all cases, &lt;math&gt;x\in (A\cap B)^c&lt;/math&gt;.

''Reason'': Proof by cases, from (15.) and (16. to 20.) and (21. to 22.).
----
{{small|(24.)}} Therefore &lt;math&gt;A^c\cup B^c\subseteq (A\cap B)^c&lt;/math&gt;.

''Reason'': From (14.) to (23.) by definition of subsets.
----
{{small|(25.)}} Therefore &lt;math&gt;(A\cap B)^c = A^c\cup B^c&lt;/math&gt;.

''Reason'': From (13.) and (25.) by definition of set equality.
{{robelbox/close}}</text>
      <sha1>s9u7rdvlsgq1qrmpkn8h2h4o4pquhdq</sha1>
    </revision>
    <revision>
      <id>2623021</id>
      <parentid>2623020</parentid>
      <timestamp>2024-04-26T11:39:44Z</timestamp>
      <contributor>
        <username>Addemf</username>
        <id>2922893</id>
      </contributor>
      <comment>/* Universal Set and Complement */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13927" xml:space="preserve">== Set Operations ==

=== Pairwise Union ===

[[File:PolygonsSetUnion.svg|thumb|A pictorial representation of the union of sets ''A'' and ''B''.  The pentagon is in both sets, but in the union we only need to represent it once.]]

If you think of a set as like a bucket of elements, then the union of two sets is like taking two buckets and dumping them together into a combined bucket.

Consider for example, the union of the sets 

: &lt;math&gt;X = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;Y = \{2,3,4\}&lt;/math&gt;

The union is then 

: &lt;math&gt;X\cup Y = \{1,2,3,4\}&lt;/math&gt;

Note that there is no need to repeat any duplicated elements, because as we discussed earlier, sets are defined only by membership.

{{definition|name=union|value=
If ''X'' and ''Y'' are any two sets then their '''union''' is defined as 
: &lt;math&gt; X\cup Y = \{z:z\in X\text{ or } Y\}&lt;/math&gt;
}}


{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all natural numbers divisible by 3 and ''B'' the set of all natural numbers divisible by 5.

List the three smallest elements of the set &lt;math&gt;A\cup B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Suppose that ''A'' has 3 elements and ''B'' has 6.  

If &lt;math&gt;A\subseteq B&lt;/math&gt; then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?

If ''A'' and ''B'' have no shared elements, then how many elements are in &lt;math&gt;A\cup B&lt;/math&gt;?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' is any set then what is &lt;math&gt;A\cup \emptyset&lt;/math&gt;? 
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Can we solve set equations the way that we solve algebraic equations?

Suppose that ''X'' is some set which satisfies the equation 
: &lt;math&gt;X \cup \{1,2,3\}=\{-2,1,2,3,4\}&lt;/math&gt;

Can we determine what ''X'' must be?

If we cannot infer what ''X'' is, then can we at least make some inference about the elements of ''X''?
{{robelbox/close}}

=== Pairwise Intersection ===
[[File:PolygonsSetIntersection.svg|thumb|A pictorial representation of the intersection of sets ''A'' and ''B''.]]

If the union &lt;math&gt;A\cup B&lt;/math&gt; forms the set of all elements in ''A or B'', then the intersection forms the set of all elements which are in both ''A and B''.

If &lt;math&gt;A= \{1,2,3\}, B= \{2,3,4\}&lt;/math&gt; then their intersection is 
: &lt;math&gt;A\cap B = \{2,3\}&lt;/math&gt;

{{definition|name=intersection|value=
If ''X'' and ''Y'' are sets, then their '''intersection''' is the set 
:&lt;math&gt;X\cap Y=\{z:z\in X\text{ and } z\in Y\}&lt;/math&gt;
}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be the set of all numbers divisible by 3, and ''B'' the set of all numbers divisible by 5.  

Find the first three elements of &lt;math&gt;A\cap B&lt;/math&gt;.
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let ''A'' be any set, and find &lt;math&gt;A\cap \emptyset&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Let 
: &lt;math&gt;A = \{1,2,3\}&lt;/math&gt;
: &lt;math&gt;B = \{2,3,4\}&lt;/math&gt;
: &lt;math&gt;C = \{1,2,5\}&lt;/math&gt;

Compute 
: &lt;math&gt;A\cap (B\cup C)&lt;/math&gt; 
and 
: &lt;math&gt;(A\cap B)\cup C&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the equation 
: &lt;math&gt;A\cap (B\cup C) = (A\cap B)\cup C&lt;/math&gt;
for sets ''A, B, C''.

Is this always true?  That is to say, is it true for ''every'' choice of ''A, B, C''?

Is it sometimes true?  That is to say, is it true for ''some'' choice of ''A, B, C''?
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
If ''A'' has 3 elements and ''B'' has 6, what is the maximum number of elements in &lt;math&gt;A\cap B&lt;/math&gt;?  What is the minimum number?
{{robelbox/close}}
=== Set-Difference ===
[[File:PolygonsSetDifference.svg|thumb|A pictorial representation of the relative complements, for sets ''A'' and ''B''.]]

If ''A'' and ''B'' are sets then &lt;math&gt;A\smallsetminus B&lt;/math&gt; represents the elements of ''A'', but ''removing'' the elements which are in ''B''.

If &lt;math&gt;A=\{1,2,3\}, B=\{2,3,4\}&lt;/math&gt; then &quot;''A'' set-difference ''B''&quot; is 
: &lt;math&gt;A\smallsetminus B = \{1\}&lt;/math&gt; 
and &quot;''B'' set-difference ''A''&quot; is 
: &lt;math&gt;B\smallsetminus A = \{4\}&lt;/math&gt;

{{definition|name=set-difference|value=
Let ''A'' and ''B'' be sets.  Then '''''A'' set-difference ''B''''' is defined as 
: &lt;math&gt;A\smallsetminus B = \{z:z\in A \text{ and not } z\in B\}&lt;/math&gt;
}}

{{robelbox|title=Exercise|theme=2}}
For any set ''A'', find 
: &lt;math&gt;A\smallsetminus \emptyset&lt;/math&gt;
and 
: &lt;math&gt;\emptyset \smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', find 
: &lt;math&gt;(A\smallsetminus B)\cap (B\smallsetminus A)&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
For any sets ''A'' and ''B'', consider the equations
: &lt;math&gt; A\cup B = (A\smallsetminus B)\cup B&lt;/math&gt;
: &lt;math&gt;A\cup B = (A\smallsetminus B)\cup (B\smallsetminus A)&lt;/math&gt;

For each equation, is it always true?

Is it sometimes true?
{{robelbox/close}}
=== Universal Set and Complement ===
[[File:PolygonsSetComplement.svg|thumb|A pictorial representation of the complement of a set.]]
In many settings, we will be exclusively interested in sets which are all subsets of a single set.  

For instance, in some settings we will only be interested in the natural numbers, and all of the sets that we consider will be subsets of it.

Or we might only be interested in the set of real numbers, or other kinds of sets, like the set of all matrices.

Whenever we restrict our interests to the subsets of one particular set, we call that one set the &quot;universal set&quot;.  We may choose the universal set to be any set that we like.

Once we specify a universal set, we can then define the notion of &quot;the complement of a set&quot;.

Suppose that we choose the universal set &lt;math&gt;U=\{0,1,2,3,4\}&lt;/math&gt; and consider the subset &lt;math&gt;A=\{2,3\}&lt;/math&gt;.  Then the complement of ''A'' is the set of all elements in the universe which are ''not'' in ''A''.
: &lt;math&gt;A^c = \{0,1,4\}&lt;/math&gt;

Notice that the choice of universal set will affect the complement of a given set.  If the universal set is &lt;math&gt;U=\{1,2,3,4\}&lt;/math&gt; and &lt;math&gt;A=\{2,3\}&lt;/math&gt; then 
: &lt;math&gt;A^c = \{1,4\}&lt;/math&gt;
{{robelbox|title=Exercise|theme=2}}
Show that for any universal set ''U'' and subset &lt;math&gt;A\subseteq U&lt;/math&gt;, 
: &lt;math&gt;A^c = U\smallsetminus A&lt;/math&gt;
{{robelbox/close}}
{{robelbox|title=Exercise|theme=2}}
Consider the universal set &lt;math&gt;U=\Bbb N&lt;/math&gt; and subsets &lt;math&gt;A=\{2,4,6,\dots\}&lt;/math&gt; and &lt;math&gt;B = \{1,2\}&lt;/math&gt;.

Find the following sets.

(1.) &lt;math&gt;A^c&lt;/math&gt;

(2.) &lt;math&gt;B^c&lt;/math&gt;

(3.) &lt;math&gt;A^c \cap B^c&lt;/math&gt;

(4.) &lt;math&gt;(A\cup B)^c&lt;/math&gt;
{{robelbox/close}}
We can now state an important relationship between the union, intersection, and complement.  
{{robelbox|title=Theorem: U-C De Morgan's}}
''Theorem'': Suppose the universal set is ''U'' with subsets ''A'' and ''B''.  Then 
: &lt;math&gt;(A\cup B)^c = A^c \cap B^c&lt;/math&gt;
----
In order to prove that two sets are equal, we must prove that every element in the left set is in the right set; and every element in the right set is in the left set.

In short, the fundamental way to prove a set equality, &lt;math&gt;X=Y&lt;/math&gt;, is to prove two subset relations.

: &lt;math&gt;X\subseteq Y&lt;/math&gt;
and 
: &lt;math&gt;Y\subseteq X&lt;/math&gt;
We now do just that, with ''X'' identified with the left side, &lt;math&gt;(A\cup B)^c&lt;/math&gt;.  ''Y'' is identified with the right side, &lt;math&gt;A^c\cap B^c&lt;/math&gt;.

''Proof'':

(For &lt;math&gt;(A\cup B)^c\subseteq A^c \cap B^c&lt;/math&gt;.)

{{small|(1.)}} Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.  

(The goal then is to show &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.)
----
{{small|(2.)}} So &lt;math&gt; x\notin A\cup B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (2.) and the definition of union.
----
{{small|(4.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (3.) by logical inference.
----
{{small|(5.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (4.) and the definition of complement.
----
{{small|(6.)}} So &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

''Reason'': From (5.) and the definition of intersection.  
----
{{small|(7.)}} So &lt;math&gt;(A\cup B)^c\subseteq A^c\cap B^c&lt;/math&gt;.

''Reason'': Lines (1.) to (6.) and the definition of subsets.
----
{{small|(8.)}} (For &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.)

Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

(The goal then is to show &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.
----
{{small|(9.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; and &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (8.) and the definition of intersection.
----
{{small|(10.)}} So &lt;math&gt;x\notin A&lt;/math&gt; and &lt;math&gt;x\notin B&lt;/math&gt;.

''Reason'': From (9.) and the definition of the complement.
----
{{small|(11.)}} So ''x'' is not in ''A'' or ''B''.

''Reason'': From (10.) and logical inference.
----
{{small|(12.)}} So &lt;math&gt;x\notin A\cup B&lt;/math&gt;.

''Reason'': From (11.) and the definition of union.
----
{{small|(13.)}} So &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;.

''Reason'': From (12.) and the definition of complement.
----
{{small|(14.)}} So &lt;math&gt;A^c\cap B^c\subseteq (A\cup B)^c&lt;/math&gt;.

''Reason'': From (8.) to (13.) and the definition of subsets.
----
{{small|(15.)}} So &lt;math&gt;(A\cup B)^c = A^c\cap B^c&lt;/math&gt;.

''Reason'': From (7.) and (14.) and the definition of set equality.
{{robelbox/close}}

[[File:Algebra1 ins fig017 dem.svg|thumb|A representation of the I-C De Morgan's law.  The notation &lt;math&gt;\overline A&lt;/math&gt; is an alternate notation for the complement, &lt;math&gt;A^c&lt;/math&gt;.]]

Notice that the way that this proof flows, is to start from formal statements, like &quot;Let &lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot; or &quot;Let &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.&quot;

From there, we progressively &quot;unpack&quot; this formalism, into a language which is more like a logical or natural language.  For example, from &quot;&lt;math&gt;x\in (A\cup B)^c&lt;/math&gt;&quot;, we eventually arrive at &quot;&lt;math&gt;x\notin A\cup B&lt;/math&gt;&quot; and then &quot;''x'' is not in ''A'' or ''B''&quot;.

At this point we have almost entirely exchanged the formalism for logical expressions.  Complements (formal) have all been exchanged for negations (logical).  Union (formal) has been exchange for disjunction (logical).  

Now that we have exchanged formalism for logical expression, we are free to reason logically.  We use that freedom to reason that &quot;''x'' is not in ''A'' or ''B''&quot; is logically equivalent to &quot;''x'' is not in ''A'' and ''x'' is not in ''B''&quot;.  

After this, we return everything to formalism.  At the end of this sequence, we ultimately arrive at the goal: &lt;math&gt;x\in A^c\cap B^c&lt;/math&gt;.

In the following proof, we repeat a very similar flow.  However, the logic required is a bit different from the above proof.
{{robelbox|title=Theorem: I-C De Morgan's}}
''Theorem'': Let ''A'' and ''B'' be two sets with universe ''U''.  Then 
: &lt;math&gt;(A\cap B)^c= A^c\cup B^c&lt;/math&gt;
----
As is common with set equalities, we again prove two subset relations.

''Proof'':

{{small|(1.)}}  Let &lt;math&gt;x\in (A\cap B)^c&lt;/math&gt;.
----
{{small|(2.)}} So &lt;math&gt;x\notin A\cap B&lt;/math&gt;.

''Reason'': From (1.) and the definition of complement.
----
{{small|(3.)}} So ''x'' is not in ''A'' and ''B''.

''Reason'': From (2.) and the definition of intersection.
----
{{small|(4.)}} So either ''x'' is not ''A'', or ''x'' is not in ''B''.

''Reason'': From (3.) and logical inference.
----
{{small|(5.)}} Assume ''x'' is not in ''A''.

(This is the start of a proof by cases.  In outline, we will assume ''x'' is not in ''A'' and then prove &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.  Then we will assume ''x'' is not in ''B'' and then again prove &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.  Because of (4.) we will therefore conclude that, in every case, &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.)
----
{{small|(6.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt;.

''Reason'': From (5.) and the definition of complement.
----
{{small|(7.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt; or &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (6.) and logical inference.
----
{{small|(8.)}} Then &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': From (7.) by definition of union.
----
{{small|(9.)}} Now assume &lt;math&gt;x\in B^c&lt;/math&gt;.

(We have finished the first case with line (8.) and this begins the second case.)
----
{{small|(10.)}} Then &lt;math&gt;x\in A^c&lt;/math&gt; or &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (9.) by logical inference.
----
{{small|(11.)}} Then &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': From (10.) by definition of union.
----
{{small|(12.)}} Therefore &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.

''Reason'': Proof by cases, from (4.) and (8.) and (11.).
---- 
{{small|(13.)}} So &lt;math&gt;(A\cap B)^c \subseteq A^c\cup B^c&lt;/math&gt;.

''Reason'': From (1.) through (12.) and the definition of subsets.
----
{{small|(14.)}} Let &lt;math&gt;x\in A^c\cup B^c&lt;/math&gt;.
----
{{small|(15.)}} So &lt;math&gt;x\in A^c&lt;/math&gt; or &lt;math&gt;x\in B^c&lt;/math&gt;.

''Reason'': From (14.) by definition of union.  
----
{{small|(16.)}} Assume (for proof by cases) &lt;math&gt;x\in A^c&lt;/math&gt;.
----
{{small|(17.)}} Then &lt;math&gt;x\notin A&lt;/math&gt;.

''Reason'': From (16.) by the definition of complement.
----
{{small|(18.)}} Then ''x'' not in ''A'' and ''B''.

''Reason'': From (17.) by logical inference.
----
{{small|(19.)}} Then &lt;math&gt;x\notin A\cap B&lt;/math&gt;.

''Reason'': From (18.) by definition of intersection.
----
{{small|(20.)}} Then &lt;math&gt;x\in(A\cap B)^c&lt;/math&gt;.

''Reason'': From (19.) by definition of complement.
----
{{small|(21.)}} Now assume (for the second case, in the proof by cases) that &lt;math&gt;x\in B^c&lt;/math&gt;.
----
{{small|(22.)}} Then &lt;math&gt;x\in (A\cap B)^c&lt;/math&gt;.

''Reason'': Repeat a sequence of steps, ''mutatis mutandis'', like the steps from (16.) to (19.).
----
{{small|(23.)}} Therefore, in all cases, &lt;math&gt;x\in (A\cap B)^c&lt;/math&gt;.

''Reason'': Proof by cases, from (15.) and (16. to 20.) and (21. to 22.).
----
{{small|(24.)}} Therefore &lt;math&gt;A^c\cup B^c\subseteq (A\cap B)^c&lt;/math&gt;.

''Reason'': From (14.) to (23.) by definition of subsets.
----
{{small|(25.)}} Therefore &lt;math&gt;(A\cap B)^c = A^c\cup B^c&lt;/math&gt;.

''Reason'': From (13.) and (25.) by definition of set equality.
{{robelbox/close}}</text>
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      <comment>New resource with &quot;==Also Known As== * Family name:  ==Overview==  ==Acquaintances, Friends and Enemies== ===Acquaintances=== ===Friends=== ===Enemies=== ==Organizations==  ==Timeline== '''1897 July 2''', X (#X on the list of people who attended) attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]].  ==Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball== At the Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire'...&quot;</comment>
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* Family name:

==Overview==

==Acquaintances, Friends and Enemies==
===Acquaintances===
===Friends===
===Enemies===
==Organizations==

==Timeline==
'''1897 July 2''', X (#X on the list of people who attended) attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]].

==Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball==
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]]

The Lafayette Archive seems not to contain the negative for this portrait.
Two higher resolution images of Lord Stanley in costume can be found on the Lafayette Negative Archive:

The 9th edition of the Encyclopædia Britannica offers a sense of what was generally available in 1897 to those interested in that kind of research. According to the Britannica, [H. M. S. [Stephens, H. Morse] &quot;Orleans, Dukes of.&quot; Encyclopaedia Britannica: An Dictionary of Arts, Sciences, and General Information. Ed., Thomas Spencer Baynes. 9th ed. Vol. XVII (Vol. 17): Mot to Orm. Google Books. Retrieved April 2023. http://...]

==Demographics==
* Nationality:
===Residences===
==Family==
*

===Relations===

==Memoirs, Papers, Biographies
=Personal Papers=

==Notes and Questions==
==Footnotes==
{{reflist}}</text>
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      <text bytes="2219" xml:space="preserve">==Also Known As==
* Family name: Parkinson-Fortescue
* Baron Carlingford
* Baron Clermont of Dromisken

==Overview==

==Acquaintances, Friends and Enemies==
===Acquaintances===
===Friends===
===Enemies===
==Organizations==

==Timeline==
==Demographics==
* Nationality:
===Residences===
==Family==
*Chichester Samuel Parkinson-Fortescue, 1st and last Baron Carlingford (18 January 1823 – 30 January 1898)&lt;ref name=&quot;:0&quot;&gt;&quot;Chichester Samuel Parkinson-Fortescue, 1st and last Baron Carlingford.&quot; {{Cite web|url=https://www.thepeerage.com/p1335.htm#i13341|title=Person Page|website=www.thepeerage.com|access-date=2024-04-25}} ''The Peerage: A Genealogical Survey of the Peerage of Britain as well as the Royal Families of Europe''.  https://www.thepeerage.com/p1335.htm#i13341.&lt;/ref&gt;
*Frances Elizabeth Anne Braham (4 January 1821 – 5 July 1879)&lt;ref&gt;&quot;Frances Elizabeth Anne Braham.&quot; {{Cite web|url=https://www.thepeerage.com/p1096.htm#i10951|title=Person Page|website=www.thepeerage.com|access-date=2024-04-25}} ''The Peerage: A Genealogical Survey of the Peerage of Britain as well as the Royal Families of Europe''.  https://www.thepeerage.com/p1096.htm#i10951.&lt;/ref&gt;

===Relations===

== Memoirs, Papers, Biographies ==

=== Personal Papers ===

# &quot;291. ''1838'' PARKINSON-FORTESCUE, Chichester Samuel, ''2nd Baron Carlingford'', statesman. Diary, 1838–98. Political matters, dinner parties, London social life, the weather. Touching description of his grief at his wife's death in 1879. BL Add MSS 63,654–704.&quot;&lt;ref&gt;&quot;Unpublished London Diaries.&quot; {{Cite web|url=https://www.british-history.ac.uk/london-record-soc/vol37/pp22-46|title=Checklist of Unpublished Diaries: nos 1-294 {{!}} British History Online|website=www.british-history.ac.uk|access-date=2024-04-25}} https://www.british-history.ac.uk/london-record-soc/vol37/pp22-46.&lt;/ref&gt; [BL might mean British Library or Bodleian Library.]

==Notes and Questions==

# Chichester Samuel Parkinson-Fortescue was Frances Braham's 4th husband.
# His titles became extinct when he died, so there wasn't a 2nd Baron Carlingford; he &quot;succeeded as the ''2nd Baron Clermont of Dromisken, co. Louth [I., 1852]'' on 29 July 1887.&quot;&lt;ref name=&quot;:0&quot; /&gt;

==Footnotes==
{{reflist}}</text>
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  <page>
    <title>User talk:Dc.samizdat/A symmetrical arrangement of 120 11-cells</title>
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      <comment>/* Building the building blocks themselves */ new hypothesis</comment>
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      <text bytes="1093" xml:space="preserve">== Building the building blocks themselves ==

And here at last with the pentad and hexad orthoschemes we must be able to find formulae describing each polytope that relate &lt;math&gt;pentads^4 = hexads^4&lt;/math&gt;, the two equivalent constructions from root systems for every 4-polytope, even for polytopes which are obviously constructed one way, and not so obviously the other. One possible construction is always by pentad orthoschemes, since it is only necessary to construct the polytope's characteristic 4-orthoscheme, and 4-orthoschemes are pentads themselves. The other is by the hexads of the 16-cell, since 16-cells compound into everything larger. Surely every uniform 4-polytope can be constructed by some function of either of these root systems, however indirect the recipe. The expressions to do so then, with an equal sign between them, make a conservation law defining the 4-polytope, which we may call its physics by Noether's theorem. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 07:11, 26 April 2024 (UTC)</text>
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  <page>
    <title>Equirectangular projection/Maps and Distortion</title>
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        <username>Bert Niehaus</username>
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      <comment>New resource with &quot;== Equirectangular Projection for Maps == The following image shows an equirectangular projection of the world. The standard parallel is the equator (plate carrée projection) [[File:Equirectangular projection SW.jpg|center|300px|upright=1.75|Equirectangular projection of the world; the standard parallel is the equator (plate carrée projection).]]  == Distortion == Equirectangular projection with [[w:en:Tissot's indicatrix|Tissot's indicatrix]] of deformation and with t...&quot;</comment>
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      <text bytes="2099" xml:space="preserve">== Equirectangular Projection for Maps ==
The following image shows an equirectangular projection of the world. The standard parallel is the equator (plate carrée projection)
[[File:Equirectangular projection SW.jpg|center|300px|upright=1.75|Equirectangular projection of the world; the standard parallel is the equator (plate carrée projection).]]

== Distortion ==
Equirectangular projection with [[w:en:Tissot's indicatrix|Tissot's indicatrix]] of deformation and with the standard parallels lying on the equator. The deformation of a circle into an [[w:en:Ellipse|ellipse]] is visible on different location the world map. 
 
=== Deformation of a Circle - Distortion Indicator ===
The deformation of the circle is an indicator for the distortion of the image.
[[File:Plate Carrée with Tissot's Indicatrices of Distortion.svg|center|300px|upright=1.75|Equirectangular projection with Tissot's indicatrix of deformation and with the standard parallels lying on the equator]]

=== Areas of Interest ===
In the image above the distortion in the planar projection is
* minimal close to the equator and
* maximal at the south pole and north pole.
Rotating circle over the equator (e.g. intersecting with North and South Pole) can be used to have projections with minimal distortion in the area of interest.

=== North Pole and South Pole ===
The strongest distortion can be found at the north pole and south pole in the following true-colour satellite image of the earth. In the equirectangular projection the top horizontal line of pixels represent the single pixel for the north pole on the sphere model of the earth. Similar to that the  south pole as one pixel is stretched out bottom line of pixels in the equirectangular projection.

== Learning Activities ==
* Explain why a projection of a sphere to a two dimensional plane can preserve some parameters can be kept and some parameters are transformed and create a distortion.
* What are the benefits and constraints of an equirectangular projection (e.g. in the context of navigation of vehicles? 

== See also ==
* [[Collaborative Mapping]]</text>
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    <revision>
      <id>2623013</id>
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      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
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      <minor />
      <comment>added [[Category:Mapping]] using [[Help:Gadget-HotCat|HotCat]]</comment>
      <model>wikitext</model>
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      <text bytes="2121" xml:space="preserve">== Equirectangular Projection for Maps ==
The following image shows an equirectangular projection of the world. The standard parallel is the equator (plate carrée projection)
[[File:Equirectangular projection SW.jpg|center|300px|upright=1.75|Equirectangular projection of the world; the standard parallel is the equator (plate carrée projection).]]

== Distortion ==
Equirectangular projection with [[w:en:Tissot's indicatrix|Tissot's indicatrix]] of deformation and with the standard parallels lying on the equator. The deformation of a circle into an [[w:en:Ellipse|ellipse]] is visible on different location the world map. 
 
=== Deformation of a Circle - Distortion Indicator ===
The deformation of the circle is an indicator for the distortion of the image.
[[File:Plate Carrée with Tissot's Indicatrices of Distortion.svg|center|300px|upright=1.75|Equirectangular projection with Tissot's indicatrix of deformation and with the standard parallels lying on the equator]]

=== Areas of Interest ===
In the image above the distortion in the planar projection is
* minimal close to the equator and
* maximal at the south pole and north pole.
Rotating circle over the equator (e.g. intersecting with North and South Pole) can be used to have projections with minimal distortion in the area of interest.

=== North Pole and South Pole ===
The strongest distortion can be found at the north pole and south pole in the following true-colour satellite image of the earth. In the equirectangular projection the top horizontal line of pixels represent the single pixel for the north pole on the sphere model of the earth. Similar to that the  south pole as one pixel is stretched out bottom line of pixels in the equirectangular projection.

== Learning Activities ==
* Explain why a projection of a sphere to a two dimensional plane can preserve some parameters can be kept and some parameters are transformed and create a distortion.
* What are the benefits and constraints of an equirectangular projection (e.g. in the context of navigation of vehicles? 

== See also ==
* [[Collaborative Mapping]]

[[Category:Mapping]]</text>
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